Sit-ups: those things sporty people do - from lying down to a full sitting position! Often done with the feet trapped. Can be done with VERY little effort, because trapping the feet gives you something to pull on and the LEGS do most of the work.
They also tend to get done very quickly - so your body weight gets the work done, as you bounce up and down using gravity and that 'bounce' off the floor to get the move completed.
Crunches: work by moving the heaviest part of the body (your head and torso) through a small(er) range of movement, using an isolated groups of muscles.
I won't go into the science - cos people like to talk about 'upper' and 'lower' abs - cos that's what you FEEL as you do crunches, rather than the way the muscles themselves work (sorry, I am a sports lecturer and am winding myself up for a return to College) BUT by moving more slowly you recruit more muscle fibres and more muscle groups to complete the work. By adding a twisting movement you can increase the usege of the deeper abdominal muscles (internal and external obliques - or the outside edges of that six pack thing) and by adding more control and postural alignment you can add the deepest muscles around the pelvic girdle - the transversus - for increased stability and support.
For sports or activity specific fitness the sit ups can be very effective.
For fitness and strengthening of the core then crunches are most effective - and less likely to cause injury.
As for technique, Active.com has a good written answer, that neither the sports scientist or instructor in me would disagree with
And this guy is quite sweet, he may stumble over his description and move a bit fast but his technique for a basics crunch is good
Do the basics first then add in the twists or some resistance (more later if you like) but start off slowly and build up your work load - you have to work out what you CAN do before you add weights or other difficulties!
It is often quite confusing to start exercise - but lots of people on here have lots of different viewpoints and experince of it - so you are bound to find someone who can explain what you need - just keep on asking