''Ja'' is a pronoun and it means ''I''.
The pronouns don't necessarily need to be used in spoken/written Slovak. Every verb has its own conjugated version for every pronoun so therefore we just skip them (the pronouns), unless, as Peachey already mentioned, we want to emphasise or stress that I am doing the action.
In Spanish, the rules about using pronouns are similar.
Hi Tristan,
As far as I can tell (read: I'm still just managing the language and hopefully other native speakers can clarify better), it comes up in a sentence when you need to specify "me, but not other people" or to emphasise that you are doing the action. I remember once ordering with a large group of friends in a Slovak restaurant and the usual "Dám si..." phrase for ordering was often replaced by "Ja si dám...", ie. "I'll have..."
Patrik is talking about cases/declensions (pády), which I guess you've already run into. I did a podcast on Slovak declensions a while back:
<em>https://slovaksnack.wordpress.com/2012/07/26/episode-18-pady/</em>