How is The Tempest considered a comedy?

The Tempest, by William Shakespeare, had its first production on November 1st, 1611, and is one of the author’s last plays. Shakespearean plays were written to fit into three distinct categories: histories, tragedies, and comedies. The Tempest is a comedy because of one main reason: there is a marriage at the end of the play.
In Elizabethan England, there was a definite conscious effort by playwrights to pen tragedies, which ended with the hero isolated or dead. Some plays are considered “problematic plays,” which have tragic themes but a marriage at the end to uplift the theme. This is why a very serious play such as Measure for Measure is also grouped into the comedy category—because there is an expressed union in the finale.
However, in The Tempest, it is not just the marriage of Ferdinand and Miranda that makes this play a comedy—there are several areas of comedic elements, characters, and scenes. Often in Shakespeare plays the broad comedic scenes are the subplots, or “B” plots, that are a secondary part of the main story and most likely to include the clownish characters. The language is therefore often written in prose, not verse, to attract the ears of the “commoners” (or groundlings, as they were called)—the patrons who paid a haypenny, stood in the yard of the theater, and were mostly uneducated folk looking to be entertained.
Shakespeare and his contemporaries consciously wrote for all classes, which is why the plays are contain both poetry (verse) and prose (regular speech).
There are several examples of this in The Tempest. Prospero, the banished magician from Milan (pronounced “millen” at the time), calls in his spirits from the island he now inhabits with the following speech:

Ye elves of hills, brooks, standing lakes and groves,And ye that on the sands with printless footDo chase the ebbing Neptune and do fly himWhen he comes back; you demi-puppets thatBy moonshine do the green sour ringlets make,Whereof the ewe not bites, and you whose pastimeIs to make midnight mushrooms, that rejoiceTo hear the solemn curfew. . . .

We can compare this with the speech of Trinculo, a clownish character and shipwreck survivor washed up on the same isle:

Here's neither bush nor shrub, to bear off any weather at all, and another storm brewing; I hear it sing i' the wind: yond same black cloud, yond huge one, looks like a foul bombard that would shed his liquor.

Comedic scenes can pepper any Shakespeare play, but to make it a true comedy, there must be a union (marriage) between characters at the end!

When it was published in the First Folio edition of Shakespeare's plays, The Tempest was grouped together with the comedies. Although the comedy here is on the whole more subtle and understated than in, say, Much Ado About Nothing, it's there all the same, and it brings some much-needed comic relief to the proceedings.
One such example comes in the form of Caliban's shameless sucking-up to Stephano. Caliban is so desperate to come out from underneath Prospero's thumb that he's prepared to prostate himself at the feet of a drunken butler. In other words, Caliban wants to become the servant of a servant, and a drunken one at that. As part of his attempt to ingratiate himself with Stephano Caliban shamelessly offers to kiss his feet.
As with much of the humor in The Tempest, this is comedy tinged with more than a hint of sadness. For we can only feel pity for poor old Caliban: he's so anxious to escape Prospero's control that he chooses to degrade himself before such an apparently worthless nonentity.
Stephano provides additional humor, albeit of a much broader variety, earlier in the scene. In his drunken state, he mistakes Caliban and the jester Trinculo for a four-legged creature. The fool is hiding beneath Caliban's cloak to escape from a raging thunderstorm. Stephano observes that this strange creature appears to be able to talk out of its behind as well as its head:

Four legs and two voices—a most delicate monster. His forward voice now is to speak well of his friend. His backward voice is to utter foul speeches and to detract (Act II Scene ii).

This is low humor at its very lowest, with Stephano making a rather naughty joke about breaking wind.