Birch pruning is trickier than most hardwood species, becuase the trees have a tendency to bleed massive amounts of sap if pruned in late winter or spring, the way many trees can be safely pruned. This can harm the tree, and can leave them vulnerable to damaging insect infestations. The best time to prune birches is usually in late summer or early fall. This timing both avoids the highest sap flow, and the egg-laying season for the insects that are most damaging to birches.
As for the "weeping branches", this is a feature of silver birch, whose scientific name is Betula pendula, "pendula" referencing the hanging nature of the branches. However, this characteristic varies from individual to individual, and depends both on the genetics of the tree and on the site. It could be that your particularly tree is unusually prone to this growth habit. But it also could have to do with conditions.
Silver birches love light; they are fairly shade-intolerant even compared to other birches. As such their branches will seek light wherever it is available. If there is abundant light reaching lower levels of the tree, the leaves will grow longer.
If they're grown among other trees, they will self-prune and you will find few branches reaching the ground, mainly because the lower branches don't get enough light to justify the tree sustaining them.
How to prune? I think the best way to prune most birches is to leave them alone as much as possible, and only remove branches that are too close to something, like a building or a path. If you want the tree to have a single-trunk habit, you can prune off side-branches. But in general, birches sometimes grow with a multi-trunk habit. Silver birch is a lot less prone to such a growth habit (and more likely to grow with a single trunk) than some other birch species (such as gray birch, Betula papyrifera, a native of North America) but it still sometimes grows this way, and if it wants to grow this way, you will have a lot of work cut out for you if you want to change its growth habit. Also, when grown on open, sunny sites, silver birch sometimes likes to branch lower down on the tree. Again, you can sometimes get it to grow in a single-trunk habit by pruning off side branches.
It is always best not to prune off too much at once. If you have selected one of several trunks or branches that you want to make the main trunk, but they are all relatively equal in size, you don't necessarily want to cut off too much in one year. Instead, trim off a portion, maybe a third to a half the growth on the side branches. Then, over the next 1-3 years, continue trimming back the unwanted trunk or branches. This will minimize the stress on the tree, and allow it to invest more growth into the main trunk or branch. Usually, this approach is sufficient to get the tree to adopt a single-trunk habit.
If you pruned off too much in a previous year, you might want to go easy on it this year. Does the tree look stressed, or does it look healthy or vigorous? I would lay off the pruning if it looks visibly stressed, i.e. less healthy or vigorous than the other trees.
Lastly, look at the lighting conditions. It could be that the tree is growing in a more one-sided way because it is more shaded on one side. If that's the case, there's not much you can do about that. Either open up more light for it by pruning away a branch or two from a nearby tree that is shading it, or just accept the asymmetry. If you have a shaded site, to where the birch isn't getting enough light, you might be better off looking for a more shade-tolerant species in the long-run. In the wild, silver birch are usually replaced by more shade-tolerant trees over time.
I'm not sure where you're located so I can't give any specific recommendations, but I always recommend choosing plants native to your local area. (Silver birch is native to a wide range across Europe and north Asia and depending on how you classify birch species, possibly into northwestern North America.)
