Hi Mary and thanks for your question about pruning your Japanese Maple. First, you should wait until late winter or very early spring when the tree is dormant and the pruning wounds will not get infected.

Next, according to Master Gardener Chris Rusch:

Avoid causing stress or stimulating unsightly growth, never remove more than one-fifth of a Japanese maple’s crown; you should also not prune a branch that is more than half the diameter of the parent stem.
In addition, don’t remove more than a quarter of the foliage of any given branch. Each branch is fed by its leaves through photosynthesis. Removing too much of the foliage will starve the tree of nutrients.

She goes on to say:

• Always prune away any lower branches that look different or have dissimilar leaves from the top of the tree.
• Remove the BDD (broken, dying or dead) branches. I find that it helps a ton to crawl under the tree when it has leaves and look up. You should be able to see patches of sky from underneath. If you can’t, your plant needs some serious pruning.
• People often let their weeping maples grow extremely dense, but if you look at the classically shaped maples in Japanese gardens, they actually have a fairly open canopy. Remove any branch that touches or interferes with another
  branch. When choosing which to remove, look at the overall shape.
• You don’t want any upper branches shading lower branches. Think of the frame of an umbrella. That’s kind of the idea, but of course with a much more natural shape.

It might sound kind of complicated, but start slow and if you make a mistake, the tree will grow back and you can try again!

Good luck,