The almost simple algebraic groups $G$ are classified by their *root datum* (defined [here][1]): in short, this is a quadruple $(X,X^{\ast},R,R^{\ast})$ with $X,X^{\ast}$ dual finitely generated free abelian groups and $R\subset X, R^{\ast}\subset X^{\ast}$ finite subsets (satisfying some conditions). Think of $X$ as the character group of some maximal torus in $G$ and $X$ a collection of *roots* (in particular, $R$ will define a root system once we tensor $X$ with $\mathbb{R}$). Then, $R^{\ast}\subset X^{\ast}$ are the corresponding *co*-objects (cocharacters, coroots). This datum characterises almost simple algebraic  groups in the following sense: the *type of $G$* is determined by the root system determined by $R\subset X$ (type ABCDEFG, as you've mentioned) and the extra information supplied by knowing the co-objects determines the weight lattice of this root system. The *simply connected* groups are those groups for which the weight lattice of the root system of $G$ is equal to $X$; this is the same as those groups of each type with the `largest' (finite) centre. The *simple* groups are those groups for which $\mathbb{Z}R = X$, that is, when the root lattice equals the character lattice. 

Some nice, introductory notes are available [here][2]. Also, see Chevalley's Collected Works [here][3].

I suppose this does not completely answer your question (I haven't given a list of the simply connected simple groups) but hopefully this will give you an idea of the general framework.

EDIT: I appear to have been too slow! As I was off looking for the required list @Jay Taylor has given a complete answer. 


  [1]: http://en.wikipedia.org/wiki/Root_datum
  [2]: http://pages.uoregon.edu/brundan/math681fall07/notes9.pdf
  [3]: http://books.google.com/books/about/Classification_des_Groupes_Alg%25C3%25A9briques.html?id=LAXvAAAAMAAJ