Only the government (e.g., federal, state, local) may bring a criminal suit.[1] But in both civil / criminal cases they are still a suit under the law seeking some form of ordered relief from the Court.
Essentially, you can think of "criminal" as a category of law and relief (e.g., prison) that is only accessible to the government.
Both the government and citizens may bring "civil" suits. Very generally, the remedies in civil suits are a court order to 1) stop doing a thing; 2) to do a thing required by law / agreement; 3) to transfer property, usually but not always money, from one party to another; or 4) to define a set of rules governing the parties going forward.
[1] There may be some obscure exception to this, but it is true for everything I've ever seen / heard of.