For those who are unaware, contract law deals with a process of people exchanging things, usually in the form of promises -- "I will mow your lawn in exchange for $30" is a contract, even if it is never written down. There are a lot of interesting ways that this law works: for example you will see reasonably often something like "I'll give you $30 to mow my lawn" / "Deal!" / "What about $20?", where the one side seems to be assuming that there is no actual contract until they both say "Deal!" -- but actually the law believes that the first person implicitly consented simply by the form of the language of the "offer" above and if the sums were larger this could hypothetically be the sort of thing that a lawsuit begins over.
The "consideration" idea is about this word "exchange" that I started off the last paragraph with, it says that contract law explicitly does not apply with things that are just given away for free. If you say "Hey after I mow my lawn tomorrow I'll mow yours", and your neighbor says "great!", and then something comes up tomorrow so that you have to cut the job short only mowing your lawn and not your neighbor's lawn, the law says that was not a contract and your neighbor cannot sue you to mow their lawn for free. How do we know it wasn't a proper contract where things were exchanged? Because we can see that you don't get anything out of it. But "get anything out of it" might get into very abstract territory: for example if you are known among your neighborhood for collecting the dead grass you've mowed and turning it into expensive compost, and that's the reasonable expectation, that could be consideration. At some point -- possibly presently, I haven't reviewed them for this post -- the Creative Commons licenses said something like "to the extent that this is a contract, the consideration for me is that I'm getting to see my work used by other people who I don't know."
This exception to contract law obviously does not apply to the terms-of-service agreements for paid services. It might hold if you are not notified about the terms of service prior to making the agreement, but US law tends to be a bit lax about the details here and usually says "well if you used the service anyway after you found out about the terms of service then you agreed to be bound by those terms." See the cases on "shrinkwrap licenses" for details.
Note also that most free-software copyright licenses and so forth are simply not contracts; that doesn't mean that they have no legal standing, but just that their legal standing isn't in contract law per se.