Ice climbing is the recreational activity of climbing ice formations such as icefalls, and frozen waterfalls. (Dry-tooling is climbing rock using the tools for ice climbing.) Depending on the slope and texture of the ice, different equipment and techniques can be employed by the climber. For example, on flat ice, almost any good hiking or mountaineering boot will usually suffice. On short, low angled slopes, one can use an ice axe to chop steps. For longer and steeper slopes or glacier travel, crampons are pretty much mandatory for a safe climb. Vertical ice climbing is done with crampons and ice axes; the climber kicks their legs to engage the front points of the crampons in the ice, and then swings the axe into the ice above their head. The strength of the ice is often surprising; even if the axe only goes in a centimeter or so it is enough to pull up on. If the climber is leading, they will need to place ice screws as protection on their way up.

Most mountaineers would only consider the last scenario true ice climbing; the less steep variations are routine aspects of winter mountaineering.

Reference

Mountaineering: The Freedom of the Hills