For climbers, the Alps are one big playground of peaks, walls, and couloirs, all with an infrastructure to support access while allowing climbers to travel light.
There are plenty of classic climbing areas with trail approaches to rock walls and walk off or rappel descents. There are also a infinite number of peaks to climb that require skills for a little of everything, from snow, to ice, to glacier, to rock to then reversing it all. The 4000-meter peaks alone offer 82 summits, many of which provide a route for all levels, from the “normal routes,” to technical climbing.
Generally, the season for climbing in the Alps runs from about the beginning of July, through about mid-September. But every year is different, the weather largely dictating the climbing conditions. For easier routes and lower elevation rock climbing the season can be much longer.
For better ice and snow climbing conditions lean toward the early side of the season, often the entire month of July. Mont Blanc is normally good by then as are the big peaks of the Zermatt area, except for the Matterhorn. The Matterhorn is normally climbable from about mid-July to early or mid-September, though some years poor weather and snow on the route can keep it out of condition longer, and shut it down for the season by the end of August.
If you want to do technical rock routes on the higher summits you might want to wait until more snow has melted. Normally the very hard high elevation rock routes are done in August, as are many of the classic mixed ridge routes.