Countries are the parties in the ICJ, which dates to the founding of the UN and includes almost every country (if not every country, period). One country has to bring an action against another (see South Africa's case against Israel). The subject matter can be criminal or civil -- I think most ICJ cases have been the latter, think territorial or fishing rights disputes.

The ICC was founded in 2002, a bunch of major countries (including the U.S., Russia, and China) aren't signatories, and it (not specific countries) prosecutes individuals for international crimes (so no civil disputes).