You explore star systems and celestial objects in them. You have a scanner, which you can upgrade just like weapons and ships etc. Atm there are 500 explorable star systems I think (it was 150 patch before and just 50 prior that patch). Most of those systems are real and with actual celestial bodies in them but they aren't marked on the Galaxy map. When you come near them you can scan them and sell that data to other players, that's the current fad. Other than that, in previous patches game mostly revolved around trading which is pretty much like Euro Truck Simulator in space or just like original Elite if you played that.
There is combat but many encounters are against AI and not actual players, because community isn't big enough for that yet. It's not easy to tell the difference, beside the fact that players can chat with you. Most encounters are random and have a little bit of back-story (mostly based on what's happening on that particular place). Some missions are about hunting pirates, sometimes they will attack you, you can watch AI fighting or even picking sides. Like Civil Wars in Skyrim. You will meet other players on space stations and outposts most of the time.
Game is still in the making and it is one huge sandbox where AI and human players are mixed, it is not human based game like EVE. It is not lobby based although you can play tutorial missions which are pretty much that, Star Citizen is currently the lobby based game. Think that limit per shard of bubble how they call it is 50 players/ships so battles can become interesting if you found yourself in some huge conflict but that is very rare so far.
Big part of what makes Elite and exploration part of it is exploring planets. That isn't implemented yet but with every patch they prepare the grounds for that to happen. Currently I think you can come fairly close to planets but can't enter atmosphere layer (for objects that have it) or anything similar. You also can't leave your ship but that will come in few months after release I presume. You can see that cockpit is fully modeled just like everything else and it will allow players to explore on feet. Pretty much like Star Citizen but with less fidelity and worse graphics but on much greater scale.