For SMPS outputs and rectifier outputs, low-impedance types are sometimes required for control loop stability and can result in lower ripple. Otherwise, it's not usually critical. In some rare occasions, using low-impedance caps can actually cause instability and oscillation. Unless you know you need a low-impedance type, often best to just go with a "generic" type; they're usually cheaper anyway.
Audio caps are designed such that the parasitics (ESR, ESL, etc.) tend to cancel each other out and give flat impedance across the audio band (DC-20kHz). The specification isn't generally worth paying for in a monitor application, but can give moderately (as in sometimes measurable using equipment, but rarely the human ear) better performance in audio applications.
I'm not a huge fan of Xicon caps, but they work fine. Nichicon is something of the "gold standard", but I'm also fond of Panasonic/ECG (I don't think Mouser sells these, but Digi-Key does, and Digi-Key no longer has a $25 minimum). Both Nichicon and Panasonic seem to take a much harder line on specs like voltage, ripple current, and temperature, while the "lesser" names seem to let things fly a little. I think this is kinda like hard drive brand preference, though. Lots of hearsay, and not much foundation in actual reality.
Nippon Chemi-Con/United Chemi-Con seem to sometimes be your only option for high-capacitance, high-voltage caps often used in rectifier outputs. They seem to work OK, but products I've designed with them have not had much "field time".
Like Kevin said, if you can fit them in, go with the 105C temp spec. They'll tend to last longer. It's usually worth the extra few cents to never have to do this again. They also tend to be bigger, so review case sizes. Mouser is famous for getting the parametric data in their search wrong on electrolytic caps, so consult the actual datasheet.
Capacitance (uF) rating must be exact in many applications. Pay some attention to tolerances: electrolytics are sold with varying tolerance ranges. +/- 20% is the typical "good and centered" range, and is appropriate in most applications. For rectifier outputs, you'll usually want to shoot for the +50%/-10% type. In some cases, it's OK to futz with capacitance some, but you need to take that on a case-by-case basis: only consider this if you simply cannot find an exact replacement or have other reason to do it.
Voltage rating, like Kevin said, is pretty straightforward. Same or higher than what you're replacing. If you're replacing something that's already been replaced, try to find a parts sheet as it may be that somebody has already overspec'd. Rarely hurts to overspec voltage and can make the cap last longer, however, higher voltage tends to mean bigger caps, so again make sure it'll fit.