On ornamental plants (not fruit) there are a few basic reasons why you should prune, best remembered as the 4 Ds: Dead, Dying, Damaged and Diseased. Next, remove crossing or rubbing branches, any that have reverted to green and for shape.
Secateurs, pruning saw
Dead wood is easiest to see and remove in summer. Cut back to healthy, pale-coloured wood.
Die-back is common after early pruning where a rogue frost can catch you out. Cut back to just above a healthy bud.
Larger dead stems should be removed with a pruning saw, working very carefully so you do not damage nearby shoots.
Green shoots on a variegated plant should be removed, as they contain more chlorophyll and are stronger. If left, they will take over and you will lose the variegation.
Taking softwood cuttings
Care of Early-Flowering Clematis