To trigger this quest you must first complete Introductions, which is a much longer, much more difficult quest that requires meeting most of the people in and around Pelican Town. Once you’ve met the last of them this quest will appear, and you’ll put your friendshipping skills to the test by giving a gift to someone. This is a tutorial for improving your standing with the NPCs.
Your task is fairly simple. All you need to do is give someone a gift. The gift doesn’t really matter so long as it’s halfway decent - a Daffodil picked off of the ground will do nicely for most people - and is given to one of the NPCs with whom you can improve your standing, aka any of the NPCs on the Social tab. There are a few who do not (Gunther at the Museum is an example), so don’t waste your time on ‘em.
Once you have an NPC and a gift, select the gift from the hotkey menu and hover the cursor over the NPC of choice to hand it to them. This will complete the quest and allow you to collect 100 gold as a reward for your good deed.