I want to change the alignment of my warlock drow in game from neutral evil to neutral good. Can you just change it? Do you have to have a monster change it? Can I make the reason because of the holy light he is exposed to, since he has a celestial patron, a unicorn?
There's no hard and fast rule
At some tables, you could just erase "NE" from your character sheet and write in "NG" while at other tables the DM will want you to undergo a form of transformation that involves some role play, and perhaps a few encounters (social, dream, combat, seeking and finding, etc).
Work with your DM to come up with a good way to make this change that fits your campaign, and that fits your table.
From your comment you note:
He said as long as I can find a reason for him to have his alignment change, I could make it happen
You have already come up with a pretty good in-story reason in your question: that of meeting your celestial patron, a unicorn, who has either directly, or through a dream or vision, communicated to you and convinced you to change your alignment to neutral good to be in harmony with the powers granted to you.
Visit the Twin Paradises of Bytopia
According to Dungeon Master's Guide p.59-60, visiting the plane of Bytopia can cause an alignment change to neutral good:
Optional Rule: Pervasive Goodwill
At the end of each long rest taken on this plane, a visitor that is neither lawful good nor neutral good must make a DC 10 Wisdom saving throw. On a failed save, the creature's alignment changes to lawful good or neutral good (whichever is closer to the creature's current alignment). The change becomes permanent if the creature doesn't leave the plane within 1d4 days.
Similar effects for other alignments occur on some other planes.
Draw the Balance card of the Deck of Many Things
This is a highly unreliable method, but if you find a Deck of Many Things and draw the Balance card, your alignment changes to its opposite:
Balance. Your mind suffers a wrenching alteration, causing your alignment to change. Lawful becomes chaotic, good becomes evil, and vice versa.
(I strongly recommend against interacting with a Deck of Many Things under most circumstances.)
Voluntarily change your alignment by asking your DM
Unlike earlier editions of D&D, 5th edition has no specific rules governing alignment change, which leaves the adjudication of this case up to the DM. Simply ask how you can change your alignment.
There is at least one example of a voluntary alignment change in the books, under the Oathbreaker paladin atonement rules (DMG p.97):
The paladin who wishes to atone must first shed his or her evil alignment and demonstrate this alignment change through words and deeds.
This suggests that it's possible for a character to voluntarily change their alignment, even if there are no general rules for it.
Alignment in 5e has very few mechanical effects or consequences, which is a departure from previous editions of D&D. There are a few magic items that have alignment restrictions, but they usually have equally-statted counterparts for other alignments, and are also usually Legendary weapons you wouldn't be likely to acquire anyways.
So there's really nothing preventing you from roleplaying your character having a change of perspective. If your character has a history of bad deeds, either in the campaign or in their backstory, you'll want to handle this more gradually, and your DM should be incentivized to have this have mechanical consequences throughout the process—maybe NPCs don't buy your characters change? Maybe people still act like they're evil even though they're trying not to be that anymore?
Either way, as long as your DM is okay with it, alignment change is not really a big deal in 5e.
The Player's Handbook states (p.122, emphasis mine):
For many thinking creatures, alignment is a moral choice. Humans, dwarves, elves, and other humanoid races can choose whether to follow the paths of good or evil, law or chaos. According to myth, the good-aligned gods who created these races gave them free will to choose their moral paths, knowing that good without free will is slavery.
Therefore your character can simply "See the light" (or dark) and choose to follow a different alignment if you wish.
Different tables have different attitudes towards alignment. There are basically two schools (well, three if you count the "alignments are an oversimplification of character psyche leading to boring stereotypes; let's ignore them and put some actual thought into the motivations and value systems of our characters" school):
- The alignment dictates behavior school. Characters are supposed to behave according to their character alignment. Acting contrary to one's character alignment is considered bad roleplaying. Some tables even go so far to allow the DM to punish alignment-contradicting behavior or even allow them to forbid such actions. The only way to change alignment in such a game is due to external influences. This way of handling alignment is usually befitting for a more mechanics-driven playstyle.
- The behavior dictates alignment school. Characters have the freedom to do what they want, but you should regularly re-evaluate whether a character's actions still match their alignment. When the group notices that a character doesn't really behave like their alignment says, then they will agree that the character's personality changed and adjust the alignment on the character sheet accordingly. This way of handling alignment is usually befitting for a more roleplay-driven playstyle.
In the past decades, the attitude suggested by the DnD rules drifted from tending towards the first school and more to the second. In fact, the rules from the 5e core material make almost no references to character alignment anymore, indicating that they want to distance themselves from this once iconic personality system. But do not let that stop you from handling alignment the way you want.