Some basic terminology
A: An algorithm is a method, or recipe for how to do/make something. In Software, an algorithm is a set of rules or logical steps to be followed to go from a problem to its solution.
A: The Hamiltonian Cycle problem asks whether a given graph contains a Hamiltonian Cycle.
A: In Graph Theory, a graph is a mathematical model representing pairwise relationships between mathematical objects. For example, let's consider a road map. The road map would have any number of cities, and then some roads connecting the cities. Each of these cities corresponds to a vertex in the graph. Each road corresponds to an edge in the graph. Note that the vertices have no fixed layout to arrange them on a plane, and the edges have no geometrical properties. We could very well be describing airports and flights. In Graph Theory, we are interested in pairwise properties between the airports; meaning, given two airports, is there a direct flight between them? If so, how long does the flight take? And how much will it cost to get from airport ABC to airport DEF?
A: To quote from WolframAlpha:
"A Hamiltonian cycle, also called a Hamiltonian circuit, Hamilton cycle, or Hamilton circuit, is a graph cycle (i.e., closed loop) through a graph that visits each node exactly once. A graph possessing a Hamiltonian cycle is said to be a Hamiltonian graph."
In other words, if starting out from city X, it is possible to visit all cities in the map ( visiting each city once, and only once ), and return to city X, then the graph is said to be Hamiltonian.
A: The Hamiltonian cycle is named after Sir William Rowan Hamilton, who devised a puzzle in which such a path along the polyhedron edges of an dodecahedron was sought (the Icosian game).