what is the difference between a cycle and a circuit in graph theory? 
A cycle of a graph G, also called a circuit if the first vertex is not specified, is a subset of the edge set of G that forms a path such that the first node of the path corresponds to the last.

This is the definition of cycle in the WolframMathWorld page. I don't understand how a cycle becomes a circuit if the 1st vertex is not specified.
I thought :
a circuit is a closed walk with no repeated EDGES.
a cycle is a closed walk with no repeated edges and no repeated vertices
 A: In graph theory conventions unfortunately differ in different contexts and with different authors. Your definition of cycle is the usual one except that cycles are often given with a specific order of vertex, edge, vertex, ... . Circuits can then be considered to be cycles but with no specific starting point. 
A: Terminology aside, the important thing is that you're aware of (and can distinguish between) the following concepts:


*

*A sequence of vertices $(v_1, v_2, \dots, v_k)$ such that $v_i v_{i+1}$ for each $i$ and $v_k v_1$ are edges of the graph.


*

*Additionally, we can require the edges to be distinct...

*...or the vertices and the edges to be distinct. (Asking for the vertices but not the edges to be distinct is weird, since the only additional object it allows is the two-edge walk from $v$ to $w$ back to $v$.)


*A subgraph of the graph isomorphic to the cycle graph $C_k$ for some $k$.


All of these are occasionally called "cycles", and I would not be surprised if any of them except the last were called "circuits". You have to figure out what a cycle is from context. 
MathWorld's definition "a subset of the edge set such that ..." is essentially equivalent to my last bullet point, and is useful for talking about unicyclic graphs, for example. (Although, if we say that cycles specify the first vertex and circuits don't, then unicyclic graphs would have to be defined as "graphs with exactly one circuit", showing that even MathWorld taken on its own is not entirely consistent about this usage.)
