If you are close enough friends with a villager, that villager might give you his or her picture! The picture is a piece of furniture that you can put into your house. Every animal has its own special message that it writes on the picture, which you can see if you touch the picture while it is somewhere in your house.
For a villager to give you his or her picture, you have to be really good friends with that villager. Try to send letters with attached gifts to your villager frequently. Talk to the villager a lot, and always agree to run errands for the villager, and help the villager satisfy his or her hobby.
A villager that is walking outside might give you his or her picture, if your friendship is strong enough. The way that it works is, if the villager is walking outside and sees you, you should go to some area of town where the villager can't see you for about fifteen minutes (just walk behind a building that isn't near where the villager is walking).
If your villager has seen you outside already, but has not seen you in a while, then if you go up to that villager after staying away for about 15 minutes, the villager will look surprised (with three white lines appearing above its head) and will run up to you, looking like he or she wants to talk to you. The villager MIGHT give you his or her picture then. But the villager might not. It's random. If the villager talks about sending letters, or shows you a letter, then you might want to send some more letters to that villager to increase your chance of getting the villager picture.
In male villagers who have the lazy personality (they are the ones who talk about food a lot), there is a bug where the lazy villager will come up to you and say some things that make it sound like he think that he already gave you his picture, but he never has. This means that you will not be able to get that villager's picture with the character currently talking to him, unless that villager moves away and then comes back someday. This is a known bug, and the only way around it is to play the game as a different character and try to build up friendship with that lazy villager and hope that the villager doesn't forget to give you the picture!