Looks like a Kilmarnock Willow to me, Salix caprea 'Kilmarnock'. You can prune the branches back to the length they are currently in autumn or allow them to grow right down to the ground, which ours does. Don't prune to severely as it will end up looking very odd!

If you want catkins on it, then pruning is done after this years flowering. They flower next year on wood produced this year. So if you prune in Autumn you are removing the flowering branches.

Good point Owdb. I'd forgotten that - we rarely prune ours.

I remove any dead branches and about a third of the old ones,that way the whole top is renewed every three years.

We wait until the tree is starting to cross the road and then cut back. Not advice for most folk!