Determining which Filter from a Z-Plane Plots?
just to add a little to Matt's answer.  if you know where your poles, $p_n$, and zeros, $q_m$, are your transfer function is

$$ H(z) = H_0 \frac{(z-q_1)(z-q_2)(z-q_3)...(z-q_M)}{(z-p_1)(z-p_2)(z-p_3)...(z-p_N)} \quad \quad M \le N $$

the complex frequency response (which contains both magnitude and phase information) is

$$ H(e^{j\omega}) = H_0 \frac{(e^{j\omega}-q_1)(e^{j\omega}-q_2)(e^{j\omega}-q_3)...(e^{j\omega}-q_M)}{(e^{j\omega}-p_1)(e^{j\omega}-p_2)(e^{j\omega}-p_3)...(e^{j\omega}-p_N)} $$

and the magnitude response (which is where you determine if it's a LPF or BPF or HPF or something else like that) is:

$$ |H(e^{j\omega})| = |H_0| \frac{|e^{j\omega}-q_1| \ |e^{j\omega}-q_2| \ |e^{j\omega}-q_3|...|e^{j\omega}-q_M|}{|e^{j\omega}-p_1| \ |e^{j\omega}-p_2| \ |e^{j\omega}-p_3|...|e^{j\omega}-p_N|} $$

and $e^{j\omega}$ is a point on the unit circle.  what this means is, as your frequency increases from DC (or $\omega=0$), that corresponds to moving along the unit circle from $z=e^{j0}=1$ to other points on the unit circle.  then as your point on the unit circle gets close to a zero $q_m$, a factor of $|e^{j\omega}-q_m|$ gets small.  $|e^{j\omega}-q_m|$ is literally the distance your point on the unit circle is from that particular zero.

likewise for the poles, but your poles are in the denominator, so as your point on the unit circle gets close to a pole $p_n$, a factor of $|e^{j\omega}-p_n|$ gets small, but since you're dividing by that small number, the effect on the gain magnitude is to increase it.

so a LPF will have poles closer to the DC point of $z=e^{j0}=1$ and, perhaps, zeros closer to Nyquist $z=e^{j\pi}=-1$ so when you're close to DC you're multiplying by big numbers and when you're close to Nyquist, you're multiplying by small numbers.

likewise for a HPF, you'll have poles closer to Nyquist and zeros closer to DC.

for a BPF, you might have zeros close to both DC and Nyquist and poles close to the unit circle at other places where they will boost the gain for those bandpass frequencies.

that's how you look at the pole-zero plots on the $z$-plane to judge if it's a LPF or whatever.