Game Theory: finding nash equilibria of an extensive form game (game tree) How do you find the pure Nash equilibria from this commitment game?

The SPNE is (U, ps)

Do you find the NE by finding the best response (BR) of a player to a specific strategy of the other player?

e.g. 

BR(p)={D}
BR(q)={U}
BR(r)={D}
BR(r)={U}

BR(U)={p}
BR(D)={s}
For the SPNE, you want to proceed by backward induction as explained by Lee Sin. For all other NE you want to construct the normal form representation (the usual table for simultaneous games) and solve for NE as if players were choosing their actions simultaneously. Note that the actions of player 1 are $\{U,D\}$, while for player 2 are $\{pp, pq, pr, ps, qp, qq, qr, qs, rp, rq, rr, rs, sp, sq, sr, ss\}$. This will indeed be a very large table.

Another way to do it is simply to look for fixed points directly. For example, assume that Player 2 plays $pp$, then the best response for player 1 is $D$ but if player one chooses $D$, the best response for player 2 is either of the following actions $\{ps, qs, rs, ss\}$ since $pp$ is not in this set, it is not a fixed point. iterate that process for all the actions of player 2.  Try to convince yourself that there is no other NE, except for the SPNE. And think about how would you change the payoffs so that the SPNE is the same, but now there are other NE's. (Hint: would changing the payoff of the terminal node after U,p from (2,1) to (1,1) work?)