"The man climbed the mountain."
Translation:Mannen klättrade upp på berget.
I actually had to go look up what a "fjäll" was, so I may be entirely off-base, but here goes: a fjäll is often but not always a mountain, whereas a berg is always a mountain. A fjäll is defined specifically as a high and barren landscape feature, so it also includes relatively flat areas above the treeline.
From Swedish Wikipedia: Ett fjäll är ett berg eller ett mer flackt landskap, vars högsta punkt når över skogsgränsen. "A fjäll is a mountain or a more flat landscape, whose highest point is above the tree line."
As you can see, the sentence here has an upp as well. 'att klättra upp på ett berg' is like climbing 'on top of' the mountain, or 'up onto the mountain' – it means you reached if not necessarily the summit of the mountain, at least a high point.
klättra i berg is more general, like 'mountain climbing' – it doesn't matter how far you get, this expression is more about the process of climbing. Since it's more general, i is more often used with indefinite forms than with the definite. It's more suitable for saying I like rock climbing than I climbed that specific mountain.
'klättra på berget' is currently not in any accepted solution here and it probably shouldn't be. It might not be totally wrong, but it sounds a bit like 'climbed on the mountain' in a weird way.
Those little grammatical particles are always the hardest thing to translate, but generally speaking, you might want to think of på as "on" and i as "in". So, using Arnauti's examples, att klättra upp på ett berg would be similar to the English "to climb up on a mountain," whereas klättra i berg would be more like "climbing in the mountains."