Well, puberty means a lot of things. It means getting pubic and underarm hair, deepening voice (for boys), the body becoming sexually mature, breasts developing (for girls), etc. Hormones are also mentioned in a jokey way, but they're responsbile for most of the changes happening in your body.
Something less talked about is your emotional health - it can be a big adjustment to go through puberty, especially with the strains of relationships, possible pressure to have sex, etc.
Because so many things are involved, there's no specific cut off date. In fact, some development continues beyond what would normally be called puberty. For example, a woman's breasts might continue to develop until she's 18 - or older!
Generally, if you've got your period that's a big sign for girls. It means you are entering the next stage in your body's development, and many people would say this means puberty is well underway.
You are the only person who will know how developed your body is, as well as how you are reacting to the growth and changes.