How the Names Originated
Welf vs. Waiblingen

They originated in the 12th century from the names of rival German houses in their struggle for the title of Holy Roman Emperor. The election, favored by the Pope, of Lothair II (c. 1070–1137), Holy Roman emperor from 1133 and German king from 1125, was opposed by the Hohenstaufen family of princes. 
This was the start of the feud between the house of Welf (Guelph), the followers of the dukes of Saxony and Bavaria (Henry the Proud, 1108–1139; later of his son Henry the Lion, 1129–1195), and that of the lords of Hohenstaufen whose castle at Waiblingen  (near present-day Stuttgart) gave the Ghibellines their name. Eventually the Guelph-Ghibelline conflict gave way to a civil war which was finally settled in 1152 by the election of Frederick I (Barbarossa), the son of a Hohenstaufen father and a Welf mother.
(When Henry the Lion, (Welf, incurred the disfavor of the Holy Roman emperor Frederick Barbarossa, Waiblingen, in 1180, his lands were forfeited to a duke of the Wittelsbach family – a dynasty that was to dominate Bavarian history until the end of World War I.)
The Guelph-Ghibelline continued for another two centuries as it became a specifically Italian conflict between forces opposed to the papacy and those supporting it. In 1334 Pope Benedict XII forbade, under pain of the censures of the church, the further use of the Guelph and Ghibelline names. As late as the end of the XV century and in the beginning of the XVI, after the descent of Charles VIII into Italy,  the supporters of the French king called themselves guelphs and the opponents ghibellines.

The Hohenstaufen
The house of Hohenstaufen supplied five Holy Roman emperors:
Conrad II (1138–52) Extended his rule into Lombardy and Burgundy.
Frederick I 'Barbarossa' (1152–89) Conrad's nephew
Henry VI 'The Severe' (1190–97) Frederick Barbarossa's son
Frederick II 'Stupor Mundi'(1212–50) Frederick Barbarossa's grandson. The greatest of the dynasty, almost succeeded in unifying Italy. Responsible for giving impetus to the birth of a national language in Italy with his 'Scuola siciliana', and for giving similar support to the birth of poetry in Germany through the minnesinger Walther von der Vogelweide.
Conrad IV 'Corradino' (1250–54) Grandson of Frederick II, was defeated at Tagliacozzo in Abruzzi by Charles of Anjou and later decapitated in Naples in 1266. With him also died all hopes of a unified Italy under an emperor. The sons of Frederick II died or were jailed. Manfred died on Feb. 26,1266 and Re Enzo lived out the reaming years jailed in Bologna. The dynasty died out 1268.

In Italy

In Italy, the terms Guelfi and Ghibellini were introduced about 1242 in Florence. The names seem to have been grafted on to pre-existing papal and imperial factions within the city-republics. Eventually the original "party platforms" became obfuscated by more struggles for power by local factions so that if a rival city became Guelph, the other automatically became Ghibelline to maintain its independence.
Aribert (died 1045), Archbishop of Milan 1018–45, should have been a Guelph on the side of the Pope; instead he was one of the early leaders of the Ghibelline party. In fact 1026 he crowned the emperor Conrad II as king of Milan.
The Colonna family in Rome, an old and illustrious Italian family that produced popes, and cardinals, belonged to the Ghibelline party.

Guelphs
The Guelphs early became associated with the papacy because of their mutual Hohenstaufen enemy. They were represented by the more democratic 'middle classes' and  'merchant class' who desired a constitutional government. They represented an indigenous Italian stock and looked to the Pope for help against the Ghibellines. However this distinction became more and more blurred as we shall see in Dante's case.
The Lombard league, an association of northern Italian towns and cities (not all of which were in Lombardy) established 1164 to maintain their independence against the Holy Roman emperors' claims of sovereignty. Venice, Padua, Brescia, Milan, and Mantua were among the founders. Supported by Milan and Pope Alexander III (1105–1181), the league defeated Frederick Barbarossa at Legnano in northern Italy 1179 and effectively resisted Otto IV (1175–1218) and Frederick II, becoming the most powerful champion of the Guelph cause. Internal rivalries led to its dissolution 1250.
Brunetto Latini (c. 1220–1294) Italian man of letters and public affairs. He was attached to the Guelph party and held some of the most important offices in the republic. His most noted work is an encyclopaedia, Li Livres dou trésor, written in French. He was also the author of a didactic and allegorical poem, Il tesoretto; a moral epistle, Il favolello; a treatise on rhetoric.

Ghibellines
Aristocratic, contemptuous of the church, supported the emperor.
Ezzelino da Romano (died 1259), was a leader of the Ghibelline movement. His reputation for cruelty led to him being called 'the tyrant' and he was depicted as a tyrant in Dante's Inferno.
Guido Cavalcanti (c. 1255–1300) Arguably the greatest Italian poet before Dante. He was a friend of Dante and a leading exponent of the dolce stil nuovo. Cavalcanti married Beatrice, daughter of Farinata degli Uberti, head of the Ghibelline faction in Florence (Inferno VI, v. 79 and X, v. 22). When the leaders of both Guelphs and Ghibellines were driven out by the rulers of Florence, he was banished to Sarzana and returned to Florence only to die.
Guido Guinizelli (c. 1230–1276) was another Ghibelline of the Lambertazzi party from Bologna. He was exiled in 1274 and died never to return to his native Bologna.

In Florence

The Ghibellines, with the help of Frederick II (grandson of Frederick Barbarossa) won the first round and banished the Guelphs from the city (1249). When Frederick II died in 1250, the Guelphs came to power again for 10 years. During this period Florence flourished both economically and politically. However, the fateful battle of Montaperti (1260), in which the Florentines lost to the Sienese, was to obliterate all that the merchant middle class (Guelphs) had accomplished politically. With the Guelphs responsible for the loss, the Ghibellines resumed power, restored the old institutions, and decreed the destruction of the palaces and towers and houses which the principal exponents of the Guelph party owned in the city and in the surroundings. All of Tuscany was in the hands of the Ghibellines except Lucca. For six years Florence was forced to submit to these outrages. At the Ghibelline League convention of Empoli, it was resolved that Florence itself be razed to the ground. It would have been destroyed had it not been for the fearless defense of Farinata degli Uberti who spoke vehemently in opposition saying that he would defend his native city with his own sword.

Dante was a Guelph. Why then being a Guelph, and supposedly part of the Papal Party, did he become an enemy of Pope Boniface VIII?

Dante had fought  in the battle of Campaldino(1289), in which Guelph Florence prevailed against Ghibelline Arezzo. After this event he began to take an interest in politics so that by 1295 he took an active part in it.
In 1300, in the full ferment of the Guelph split into two factions - the moderate but 'disaffected faction' (Whites- Bianchi) and than the 'pure papal extreme faction' (Blacks - Neri) - Dante, one of the six newly elected priors of the Republic belonged to the White faction opposed to Pope Boniface VIII.

Blacks and Whites

It all started in Pistoia, a few miles from Florence. Two branches of the Cancellieri family quarreled over an insignificant incident. It seems that  one nephew was chastised by an uncle for throwing a snowball. A few days later, in revenge, the nephew struck the uncle. It would have all stopped there, for the uncle thought nothing of it. However his son, Focaccia caught his cousin and cut off his hands. Then as if this were not enough, went after the father, his uncle, and killed him. A feud developed among the ruling Guelphs over the murder which resulted in the splitting into two parties: Neri (Blacks) and Bianchi (Whites). The Bianchi got their name after Bianca Cancellieri, so the opposing faction chose the color Black as its standard. Civil war broke out.
The Florentines, in trying to put an end to the bloodshed, took the leaders of the factions in Pistoia and imprisoned them  in Florence.
This had the effect of exacerbating the existing rivalry between the two Florentine families, the Cerchi (Par. XVI, v. 65 and v. 94) and the Donati (Purg. XXIV, v. 82). Though both Guelph who had fought with Dante at the battle of Campaldino, the two families took opposing views in supporting the jailed Pistoiese leaders. The Cerchi became associated with the Whites and the Donati with the Blacks. These two divisions, though unpolitical at first, assumed more and more political overtones so that the Black Guelphs remained the pure papal Guelphs, while the White Guelphs became disaffected and eventually threw in their lot with the Ghibellines.
There ensued brawls and fights so bitter that both Pope Boniface VIII and Charles of Valois were called to interfere and put an end to the fighting. Vieri de' Cerchi (White) was summoned by the Pope but refused to do anything the Pontiff asked.
It was decided then that the leaders of both factions be banished from Florence. This was a sad affair for Dante for in exercising the duty of his office saw his best friend Guido Cavalcanti  banished to Sarzana as one of the leaders. Although they were called back after a few weeks, Guido fell ill with malaria and died in August of that same year (1300). 

The Exile
In June 19 of the next year (1301) there was a vote recorded in which Dante opposed giving one hundred men for the papal forces on requisition of Pope Boniface VIII. However in September 28 of that year, in order to protest against the papal policy and possibly avert a total subjugation of Florence to Charles of Valois, the Whites sent an embassy to Rome of which Dante was a member. They waited a long time to no avail. In November 1, 1301 when the emissaries were still in Rome waiting for an audience with the Pope, Charles of Valois entered Florence under pretext of quelling the feud between the two faction. However, once in possession of the city, Charles wasted no time in letting his true intentions be known: that he favored the Black Guelphs. The latter began to consolidate their power by getting rid of their opponents.
In January 1302 Dante and four other Whites failed to appear when summoned by the Podestà to answer charges of having extorted money and made illicit gains.They were banished from Tuscany for two years, never to hold a public office again, and were ordered to repay 5,000 florins within three days or risk forfeiture of all property.
By March 10, 1302 a heavier sentence came down upon Dante and the others. They were to be burned alive should they ever be caught. 
None of the early biographers, including Giovanni Villani who belonged to the opposing faction ever believed that Dante was guilty of any wrong doing. The accusation was just a pretext. His only crime was that of being a White Guelph.  
Dante spent the remainder of his life, nearly twenty years, in exile wondering in central and northern Italy. In his Convivio Dante gives an account of the miseries he endured: "I have gone about as a beggar, showing against my will the wound of fortune... Verily I have been a ship without sails and without rudder, driven to various harbors and shores by the parching wind which blows from pinching poverty."
He died in exile in Ravenna in 1321. His daughter, Beatrice, a nun, nursed him at the end.