Across, over & through

1) on/to the other side of (line): across and over

Across and over can both be used to mean 'on or to the other side of a line, river, road, bridge, etc':

His village is just across/over the border.
See if you can jump across/over the stream.

2) high things: over preferred:

We prefer over to say 'on/to the other side of something high'.
Why are you climbing over the wall? (*not ... across the wall?)

3) on flat areas; in water: across preferred

We usually prefer across to say 'on/to the other side of a flat area or surface', or to talk about movement in water:

He walked right across the desert, (*not ... over the desert.)
Let's swim across the river, (*not ... over the river.)

But over is sometimes used in British English if there is no idea of arriving at the other side:

We often walk over the fields in the evening.

4) the adverb over:

Note that the adverb over has a wider meaning than the preposition over. You cannot say Let's swim over the river to the church, but you can say Let's swim over to the church.

5) across and through:

The difference between across and through is like the difference between on and in. Through, unlike across, is used for a movement in a three-dimensional space, with things on all sides. Compare:

We walked across the ice. (We were on the ice.)
I walked through the wood. (I was in the wood.)
We drove across the desert.
We drove through several towns.