Patty, in a square dance, four couples form a square and dance with each other through the entire dance. In a contra dance, couples generally form double lines that stretch the entire length of the hall. A large hall with lots of dancers may have as many as 3 to 5 full lines, a smaller dance may have only 1 long line of couples.
Every other couple moves in one direction while dancing with people along the way, and the other half move in the other direction while dancing with couples they encounter along the way. The end result is that each couple winds up dancing with every other couple in the line. When you and your partner eventually get to one end of the line, you turn and start back again, dancing in turn with each couple you meet in the line.
You and your partner mostly stay together as you work your way down the line dancing, but you wind up also dancing with just about everyone else in the line too. I think that makes it particularly fun. You also cover more ground, from one end of the hall to the other, as opposed to staying in one spot like the 4 couples in a square dance do. Both contra dancing and square dancing do incorporate some steps in common, such as do-si-do and swing your partner...
This contra dance video shows how people keep dancing with different other people as they work their way up or down the line:
Here is a very lively and crowded contra dance: It looks like chaos to a non-dancer, but you can see the well organized double line of couples go 'forward and back' right on time at the 0:42 mark. they know exactly what they are doing! lol
Again, another crowded lively dance: At the 0:56 time mark in this one, again at 2:00, you can again see those organized long lines of couples go forward and back. The caller stops calling the moves after a while, as the dancers get the hang of the sequence of moves. Each dance is different. I have danced in that Greenfield MA hall, and I've danced with half the folks in that video at one time or another. That particular band is terrific- they are only two young men and they play everything you hear- electronically looping each part on top of the next as they play on different instruments- live. The tension builds as they add more parts and loop them on top of what they just played a minute before. lots of energy! I do square dances occasionally, but I find contra dances to have a different feel. Hope this adds some info...or maybe just confuses it more for you! 
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Those irritated by grain of sand best avoid beach.
-Strumelia proverb c.1990
) we did a "quadrille." It reminded me of scenes from Jane Austen movies. We were one person short of two group of eight; I suggested calling someone in from the street, but instead they used a large balloon of Mickey Mouse as a place holder (but he couldn't go round the circle weaving between the "follows," aka the ladies. Actually I think Mickey was a follow, he would have needed to weave between the leads). 
That's a major dance.
I also like to put some electrolytes powder in my BIG cold thermos of water. 
