walk

Finite or infinite sequence of edges which joins a sequence of vertexes

Closed walk → walk in which first = last vertex

trail

walk in which all edges are distinct (but not necessarily all vertexes)

path

trail in which all vertexes (and therefore also all edges) are distinct

tour

closed walk in $G$ that includes/traverses every edge at least once

cycle

non-empty trail in which the first and last vertexes are equal (but all other vertexes are distinct)

eularian trail / eularian walk

trail in a finite graph that includes/traverses every edge exactly once (but allows for revisiting vertexes)

eularian cycle / eularian tour

eularian trail that starts and ends at the same vertex

hamiltonian path

path in a graph that visits each vertex exactly once

hamiltonian cycle

hamiltonian path in which the first and last vertex are equal