Regarding "illegal moves" - if you make an illegal move, but not finish it (by pressing clock), you can still fix yourself by taking the piece away and moving it elsewhere.
Not true.
FIDE tule section 4.6:
4.6 When, as a legal move or part of a legal move, a piece has been released on a square, it
cannot be moved to another square on this move.
If you prefer USCF rules, the relevant section is 10G.
I half disagree.
@Daarzyn7 said if you make an illegal move, but the rule you quote (4.7 these days) refers to legal moves or parts of legal moves, so doesn't apply.
I say "half disagree" because 4.7 defines a move to be made only when the pieces have been released on a legal square and possibly removed from the board etc. (or at least attempts to; FIDE need to get their act together). So in terms of the laws the concept of making an illegal move - @Daarzyn7 is a non starter. It is allowed for instance to touch your pawn on on d4 with the intention of moving it, then release it on a6. The situation at that point is that you are part way through making your move. You can then move it to any number of squares (or non squares) other than d3 or d4 and the situation remains the same. But if you press the clock you have completed an illegal move.
It's not possible to make an illegal move. But you can complete one under competion rules (only).
But imagine your opponent touching 4 pieces before moving 1 ... A roller-coaster of emotions. I guess that's why they made the rule