The wikipedia entry for "non chord tones" turns out to be pretty good, with a lot of examples:
A passing note are the dissonants that are reached and left by stepwise motion in between chord tones that are typically a third or a fourth apart. They are typically short and on an unemphasized part of the beat (but not necessarily so). The "corners" of the passing notes are formed by consonants, and often they will be on accented parts of the beat:
So in this example, the e and the c in the upper voice are the chord tones that form the "corners". The d in between is the passing note. With these corners, only d is possible a passing note.
One way to think of passing notes is that they are a part of the scale to fill the gap between chord notes. In other words, they are reached by stepwise motion, and continue in that direction until encountering the next nearest chord tone.
A suspension is a "held" note. It starts off as a consonant on either an emphasized or an unemphasized part of the beat. Then the rest of the harmony moves on the strong part of the beat to something that would have been a consonant chord, save for the held or suspended note, which then gets its resolution into a consonant:
So in this example, the c in the upper voice in the 2nd measure is the suspended note. It started of as a chord note in measure 1. Then the rest of the harmony, which is in this case only the lower voice, moved on to a new chord while the upper voice keeps hanging on, forming a dissonant. This way of generating dissonants is called preparation (simply a way to say that the note that is now a dissonant, started as a consonant). Although the dissonant itself is on a strong beat, the effect is not as harsh as when it would be unprepared. Finally the supspension is resolved in beat 2 of measure to, forming a consonant again.
I'm not sure whether it is essential that a suspension is resolved by stepwise motion. However, dissonants in general tend to be resolved by step rather than by skip and I am sure this is the common case.
(This is a bit of a simplification, as there can be more than one suspended note. What is "the rest of the harmony" as opposed to "what are the suspensions" is to some extent subjective and will also depend on what chord one would normally expect at that point in the progression)
(Examples from wikipedia)