A time signature is a guideline for the rhythm of the music. It tells you the number of beats per measure and the value of each beat.
Example: In 4/4 there are 4 beats per measure and each beat is equal to 1 quarter note. In 7/8 there are 7 beats per measure and each beat is equal to 1 eight note.
In any time signature the rhythm should repeat every measure (unless you're a crazy modern composer and don't feel like following any rhythmic structure at all). Rhythms are created by series of accented and unaccented beats. The most common patterns are 2 beat and 3 beat patterns.
2 note pattern - Consists of 1 accented beat followed by 1 unaccented beat.
3 note pattern - Consists of 1 accented beat followed by 2 unaccented beats.
Example: In 4/4, beats 1 and 3 are usually accented and beats 2 and 4 aren't (two 2 beat patterns). Listen to almost any song in 4/4 (which is almost any song) and you will probably hear a drum pattern with snare hits on beats 1 and 3. In 3/4, only the 1st beat of each measure is accented (a 3 beat pattern).
Complex time signatures are made up of a combination of these patterns. They can be easily broken down into smaller beat patterns.
Example: 5/4 timing typically consists of a 2 beat pattern followed by a 3 beat pattern. That means beats 1 and 3 are accented and beats 2, 4, and 5 are not. It's also good to know that you shouldn't simplify time signatures. Say you have 8/8. Hey! It's just like 4/4! Wrong! If 8/8 is used instead of 4/4, it is for good reason. The rhythm is different (probably with beats 1, 4, and 7 accented; two 3 beat patterns followed by a 2 beat pattern) and would fit awkwardly into bars of 4/4.
Cross-rhythms are created when two different rhythms are played at the same time. A simple but effective exercise one can use to practice cross-rhythms is to tap one's foot on the 1st and 3rd beats in 4/4, while strumming the 1st and 3rd beats in 5/4. Cross-rhythms can be as simple as something in 4/4 played over something in 8/8, or more complex, like guitar switching between 5/4 and 7/4 while bass is going in 8/8 and drums in 5/4 and 7/4 (ProzaKc Blues).
Polyrhythms are different from cross-rhythms. A polyrhythm exists when two or more different rhythmic patterns exist in a single time signature.
Very interesting things can also be done by changing which beats are accented. For an example, listen to Bruford's drumming in Frame by Frame by King Crimson. In this song he plays a 4/4 pattern, but it sounds interestingly offbeat because beats 3 and 4 are accented.
Why should I care?
Complex rhythms are definitely underused and if used correctly can sound cool and not just appear to be there for the sake of complexity or pretentiousness.
In order to get a better idea of what complex rhythms sound like you can listen to these songs.
Primus Eleven (In 11/8)
Coldplay Clocks (In 8/8)
King Crimson Larks' Tongues in Aspic Part II (Mostly in 10/8)
King Crimson ProzaKc Blues (Mentioned in the cross-rhythm section)
King Crimson A great majority of their songs, especially any made in 1981 or later
Dream Theater I don't listen to them, but I know that they commonly use complex rhythms along with many other progressive metal bands.
I hope this article was relatively easy to understand as well as eye opening.
P.S. Did I mention enough Crimso songs for you?
*I doubt this is the technical term, but I'm going to use it anyways.