Partial Reps are exactly what the name says: Doing partial reps when you are no longer able to complete full range-of-motion reps. Cheating is when you use other muscles to assist the targeted muscle in lifting the weight. Now, some people believe that reps should only be done through a strict, full range-of-motion, and when you can't, you end the set- I am not one of these people, and it frustrates me when people say this. Now I am of the belief that beginners should not use either of these. A lot of beginners will do whatever they see the big guy next to them doing. So when they see him forcing out some partials or swinging back a little when he's doing his barbell curls, and veins are poppin' out of his head, he's screaming, and looks a little like he's crying too, all at the same time, they think to themselves..."hey, maybe that was what that magazine was talking about when it said intensity." So what do they do? They apply these techniques all the time, and they end up hurting themselves and/or getting disproportional muscles, because they never took the time to learn a little and discover the fact that the only way to develop a high quality physique is to work through a full-range-of-motion, and only part of the time, apply these cheating, partial rep, and other techniques used for raising the intensity.
Cheating is a way to push your muscles past momentary failure by helping to force the weight up with muscles other than the targeted ones. Now a lot of people disagree to whether or not this method works, but I can assure you that it does. Think about it: You're doing a set of barbell curls, you've done 2 sets already out of the 4 you plan to do. You got 12 reps on the first set, the next set falls to 9 reps. On this set, you're only able to get up 6, and you barely got that up, so you can do one of two things: set the weight down, or use a little bit of momentum to swing the weight up as you lift it, which allows you to get 3 more reps up. Now, if you chose choice 2, cheat some reps out, then you did do more work than if you would have set it down immediately, not less.
The key to cheating is to use only what you need. You use just enough help from other muscles so that you can keep the set going. Don't start a set swinging a weight up, and don't use this as an excuse to get more weight up. Cheating is used to help you take your muscles past failure, not make a lift easier. Cheating does works better for someone training for mass or strength, with their reps falling in the 4's, 5's, and 6's. If you're training with higher reps and geared toward good definition and quality, then I would recommend that you generally use full, strict reps. Some examples of exercises to use them on are Barbell Curls, Lateral Raises, Standing Shoulder Presses. And as I said before: beginners, start out using only full-range-of-motion reps and light weights, and when you are sure you are comfortable with the form of an exercise, you can use a little cheating if necessary.
Partials are another great way to push your muscles to a whole new level of failure. When you can't do any more full reps, just go as far as you can- three-quarters, halfway, etc. Some exercises shouldn't be anything less than halfway, like squats, for example. You can do half squats, but don't go too much less than this, and only do this at the end of your last, or maybe second to last set. On the other hand, when you perform some exercises, you can keep doing partials until you are barely moving, like Standing Calf Raises. One very important thing you must remember though: don't make a habit of ending a set early and finishing it off with partials. You must be true to yourself, don't lie to yourself by thinking you cannot get anything more when you can. And don't do a whole set of partials. Pick up a weight that you normally would, and do everything that you normally would, except finish it off with the partials.
OK now for example, say you're doing some squats and you're in your bulking phase, training for pure mass. After your warm-up, you load up the bar with what you can do for 12 reps, do them, you're done. Add 50 pounds, do 8 or 9. Add 25 more, do 6-7. Then, you stay at the same weight, or maybe add a little, and do your 6, after you complete the six, and you're shaking with fatigue, go down about halfway and come back up. Continue to drop down a little less each time, as far as you feel you can go, and then rack the bar. I know, this burns like crazy, but it works like crazy too. After that, go do some quad extensions(which you can also apply partials to on the last set), and burn those out. After that, if you've never done it before, try and walk down some steps, but be careful.
Both of these techniques are awesome ways to raise the intensity, but use them responsibly. Any questions on anything about these or training in general, e-mail me.