Half- or front-tucking is when you tuck the front but not the rest of your top - see Olivia's example above.
More often than not, it gives a casual or smart casual vibe, and thus is best done with T-shirts, shirts, tops and jumpers. Blouses are best half-tucked into trousers, not skirts.
You may also want to experiment with adding a belt for extra effect, but this is not absolutely necessary.
It is a technique I recommend for lengthening legs for many reasons:
displaying more of your bottom half obviously lengthens legs ...
... especially as a half-tuck requires roomier tops, which in turn shortens the top half visually; the combination thus offers better proportions
tucking your top only at the front creates a slight bulge at the sides, which evens out larger hips/thighs and creates a straighter, thus leaner looking figure
the extra length behind conceals bigger bums
asymmetry always helps to trick the eye when it comes to proportions
Finally, the vibe is casual and unusual, but not untidy: this is an easy way to jazz up very simple items.