Yosemite is hugely popular with climbers – the granite cliffs are massive and technically challenging and there are many, many places to climb. There are lots of options for both day climbs and multi-day climbs. If you are a non-climber watching climbers move over the face of El Capitan is quite a sight.
For more information regarding climbing in the Park go to
http://www.nps.gov/yose/planyourvisit/climbing.htm
If you are an experienced climber wanting a guide, an inexperienced climber wanting some tuition, or just fancy giving it a go there is a Mountaineering School and Guide Service in the Park.