If you say that someone is self-indulgent, you mean that they allow themselves to have or do the things that they enjoy very much.

What does it mean if someone is self-indulgent?

: excessive or unrestrained gratification of one's own appetites, desires, or whims.

What is the self-indulgent person like?

Self indulgence is often referred to someone who is being greedy, or is just generally selfish, and it is often seen as a bad thing. Some would say a self-indulgent person think about themselves a lot; they over eat, they care only about themselves, they take too much of a good thing, such as loads of cream cakes.