You can use Stargates even if your Security Status or Standings make you an "enemy of the state", but you may be fired upon depending on the circumstances (you may enter any area in a Pod without being shot, no matter what).
There are four types of Stargates (Caldari, Ammar, Minmatar, Gallente) but the only difference is in the appearance - they all function in the same way.
In the future, a feature that may be implemented is that an Alliance will be able to control Stargates in their territory (see Sovereignty). For now, anyone may use any Stargate at will (lag permitting...)
In High Security areas (0.5 to 1.0), Stargates are patrolled by Concord or the Navy of the particular faction that owns the space. There will also be 6 to 8 turrets equidistantly placed around the gate. Shooting anything you aren't supposed to will result in them opening fire. The Customs Official (in the zippy Interceptor that buzzes around) will also scan your ship if he gets close, confiscating any illegal things you possess and handing out fines if any are found.
In Low Security areas (0.1 to 0.4), Stargates are not patrolled by anything, but will will have a few turrets that will open fire on anyone who picks a fight. These turrets are not particularly powerful, and can be tanked. Sometimes there are NPC Pirates that appear.
In No Security areas (0.0), Stargates are devoid of all law enforcement, but often there will be NPC Pirates.
Note that smartbombs cannot be set off near to gates. The turrets guarding the gate have a range of 150km from the Stargate itself.
Some missions have inactive Stargate(s) in the background - they will not take you anywhere.