If you look at generic games, that is games where players are not indifferent between any choices such a classification is easy to produce. Let the strategy sets be
$$
S_1 = \left\{a,b\right\}, S_2 = \left\{c,d\right\}
$$
and let the utility/payoff functions be denoted by $U_1$ and $U_2$.
The characterization will be based on the dynamics of the game, that is given a strategy profile $(s_1,s_2)$ will any player wish to alther his strategy. This will also give you all the equilibria. Without loss of generality you can assume
$$
U_1(a,c) > U_1(b,c).
$$
(Should this not hold, switch the labels of the strategies $a$ and $b$.)
We still need to know if player 1 prefers $(a,d)$ to $(b,d)$ or vice versa, and also if player 2 prefers $(a,c)$ to $(a,d)$ and if he prefers $(b,c)$ to $(b,d)$. This gives eight possible combinations which will characterize the game classes. (But due to symmetry actually there will only be four classes.)
Category 1.
$$
U_1(a,d) > U_1(b,d) \mbox{ and } U_2(a,c) > U_2(a,d) \mbox{ and } U_2(b,c) > U_2(b,d).
$$
In this category $a$ and $c$ are strictly dominant strategies. A game like this will play out like the prisoners dilemma in that it has a single Nash-equilibrium which is also a dominant equilibrium.
Category 2.
$$
U_1(a,d) < U_1(b,d) \mbox{ and } U_2(a,c) < U_2(a,d) \mbox{ and } U_2(b,c) > U_2(b,d).
$$
This game will play out the matching pennies game, in that there are no equilibria.
Category 3.
$$
U_1(a,d) < U_1(b,d) \mbox{ and } U_2(a,c) > U_2(a,d) \mbox{ and } U_2(b,c) < U_2(b,d).
$$
This game will play out like a coordination game, e.g. Battle of the Sexes or Chicken. There are two equilibria, $(a,c)$ and $(b,d)$.
Category 4.
$$
U_1(a,d) > U_1(b,d) \mbox{ and } U_2(a,c) > U_2(a,d) \mbox{ and } U_2(b,c) < U_2(b,d).
$$
In this category $a$ is again strictly dominant. $c$ is not, but a rational player 1 will always play $a$, and $c$ is best response to that. I don't know if this game has a well accepted name, it has one Nash-equilibrium $(a,c)$.
Due to symmetry, all other combinations belong to one of these categories.
If you wish to admit indifference between several outcomes, it is still possible to use this kind of classification, but the number of categories will be much larger.
