I see frame geometry having 3 primary affects
- Fitting the rider; which you're already addressing and I won't talk about here... But a lot of geometry stuff comes down to making the other stuff work with fitting riders on the bikes. It's very important.
- Fitting stuff on the bike
- Handling characteristics.
Since you asked, I'm talking about your basic on-road diamond-frame bike. Cruiser, road, commuter, urban, touring, racer...
Fitting Stuff On the Bike
- Wheel clearance affects how big the tires can be and how easy it is to mount fenders. For the rear wheel you need to look at the chainstay bridge, seatstay bridge and seat tube. For the front wheel, look at the fork crown and downtube. Often either not measured or simply given to you as a maximum tire size measurement.
- "Toe overlap" -- how close the front wheel gets to the front of the pedals. More of an issue with fenders, and not really an issue once you're going any kind of speed. I never see this on the geometry specs from manufacturers, but definitely worth checking. You can get used to moderate toe overlap, though.
- Longer chainstay length gives you more heel clearance for loading panniers on the rear rack. I know from personal experience that the most likely place to kick a pannier off the rack is in an intersection, so if you plan to carry panniers this can be very important. Short chainstay length can be compensated for by getting a rack that allows panniers to be mounted further back (longer top, top further back, however they do it).
Handling characteristics
First off, the way a bike handles, "stability" and "maneuverability" are essentially opposites. It's a tradeoff. Whether you want stable or maneuverable depends on what you're going to do with the bike, how fast you'll be going, your riding experience, etc. And there's some more complicated harder-to-understand secondary effects.
I really think that for most of the handling stuff you're best off just test-riding bikes that fit you. Experiment with the handling. Try going slow. Try going fast. Try making a sharp turn. Try making a subtle turn. Try a dodge/weave. Try those turns at different speeds. Try a fast start. Try a fast stop. Try all that in all the handlebar positions. Unless you're designing a bike, a lot of this is all very theoretical, heavily interrelated (can't really change one variable without changing the others), and likely to be very small differences when comparing actual bikes.
Bottom Bracket (BB) height (sometimes you can find this out, sometimes you have to make a guess based on wheel+tire radius minus BB drop). Higher bottom bracket makes it easier to go over stuff. Not really an issue if you stick to roads, but can be a factor when trying to pedal through sharp turns. Higher BB also lets you use longer cranks which some people want. Compare a Cyclocross to a Road bike and the CX bike will have a higher BB.
Wheelbase length (distance between hubs/contact patches). A longer wheelbase will make the bike more longitudinally stable. In other words, less prone to wheelies and less prone to flipping you over the handlebars. This could theoretically translate to better ability to stop faster. Note that one way to give you a longer wheelbase is longer chainstays. The position of your center of gravity makes a big difference, too.
Headtube angle (steering axis angle). A shallower/slacker (more pointed forward) angle gives you more trail. A shallower angle also increases how much the weight can make the wheel flop on its own and negatively affect stability at low speed. This is usually measured from horizontal, so a steep angle will be closer to 90° and a shallow angle will be a lower number.
Fork offset/rake (how far the front hub is in front of the steering axis). More offset gives you less trail. More offset also tends to give you more toe clearance.
Trail is how far the front contact patch is behind the steering axis. Note that while the trail is the important measurement, the headtube angle and fork offset (and wheel size) are what determines the trail. A bicycle with no trail would be unrideable (but many people like having very little trail).
There's different ways of measuring the trail (along the ground, or perpendicular to the steering axis) that make comparisons more intuitive, but more trail means more stable, and less trail means more maneuverable. Or to put it another way: less trail means more twitch and more trail means less responsive.
The stability from trail increases as the bicycle goes faster. If you look at some cruiser bikes they'll tend to have a shallow headtube angle and a lot of trail: they're stable and easy to balance at low speeds but probably hard to steer at high speed. If you look at a road/racing bike, the headtube angle is usually much closer to vertical and the trail is fairly small: they may feel unstable and wobbly (hard to balance) at low speeds, but are still easy to steer (responsive) at high speeds.
Note also that the handlebar setup and rider preferences/experience make a huge difference whether your prefer maneuverable/twitchy/responsive or stable/unsteerable/unresponsive.