Are compact & connected Lie Groups in correspondence with semisimple Lie groups? I think there is a condition on the center (discrete?) but I'm not sure.

The answer to your title is "no"; lots of semisimple Lie groups are not compact (for example, $SL_2(\mathbb{R})$). You're getting this mixed up with the fact that a complex semisimple Lie group has a unique compact real form, and that this is a bijection to semisimple compact Lie groups. (Complex reductive groups are in bijection with general compact Lie groups; this allows torus factors on both sides). 


To amplify Ben's answer, I'd point to an earlier post that has lots more detail: here. The subject of compact groups is old and wellstudied, so there are many references to choose from, even Wikipedia perhaps. Anyway, it's good to browse older Lie group entries on MO first. PS: This supplementary "answer" is really a suggestion that the question is too close to the earlier post I cited to qualify as a fresh question. Textbook material of this kind calls mainly for references rather than discussion. 


I would like to add the following: I think the source of confusion is the fact that the Killing form is nondegenerate (for semisimple Lie groups) and negative definite (stronger than nondegenerate) for compact Lie groups with trivial center $SL(2)$ is semisimple but not compact. The torus $S^1$ is compact but not semisimple (abelian). Compact groups are reductive and semisimple only when in the case of trivial center. 
