(closed) walk / trek / trail / path
Graph theory^{} terminology is notoriously variable so the following definitions should be used with caution. In books, most authors define their usage at the beginning.
In a graph, multigraph^{} or even pseudograph^{} $G$,

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a walk of length $s$ is formed by a sequence of $s$ edges such that any two successive edges in the sequence share a vertex (aka node). The walk is also considered to include all the vertices (nodes) incident^{} to those edges, making it a subgraph^{}.
In the case of a simple graph (i.e. not a multigraph) it is also possible to define the walk uniquely by the vertices it visits: a walk of length $s$ is then a sequence of vertices ${\nu}_{0}$, ${\nu}_{1}$, … ${\nu}_{s}$ such that an edge ${\nu}_{i}{\nu}_{i+1}$ exists for all $$. Again the walk is considered to contain those edges as well as the vertices.

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A trek is a walk that does not backtrack, i.e. no two successive edges are the same.
For simple graphs this also implies ${\nu}_{i}\ne {\nu}_{i+2}$ for all $0\u2a7di\u2a7ds2$.

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A trail is a walk where all edges are distinct, and

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a path is one where all vertices are distinct.
The walk, etc. is said to run from ${\nu}_{\mathrm{0}}$ to ${\nu}_{s}$, to run between them, to connect them etc. The term trek was introduced by Cameron [Cam94] who notes the lexicographic mnemonic
$$\text{\mathit{p}\mathit{a}\mathit{t}\u210e\mathit{s}}\subset \text{\mathit{t}\mathit{r}\mathit{a}\mathit{i}\mathit{l}\mathit{s}}\subset \text{\mathit{t}\mathit{r}\mathit{e}\mathit{k}\mathit{s}}\subset \text{\mathit{w}\mathit{a}\mathit{l}\mathit{k}\mathit{s}}$$ 
The other terms are fairly widespread, cf. [Wil02], but beware: many authors call walks paths, and some then call paths chains. And when edges are called arcs, a trek of length $s$ sometimes goes by the name $s$arc.
Note that for the purpose of defining connectivity any of these types of wanderings can be used; if a walk exists between vertices $\mu $ and $\nu $ then there also exists a path between them. And here we must allow $s=0$ to make “are connected^{} by a path” an equivalence relation^{} on vertices (in order to define connected components^{} as its equivalence classes^{}).

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A closed walk aka circuit of length $s\ne 0$ is a walk where ${\nu}_{0}={\nu}_{s}$,

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a closed trek is a trek that’s closed in the same way, and

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a closed trail likewise;

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a closed path aka (elementary) cycle is like a path (except that we only demand that ${\nu}_{i}$ for $$ are distinct) and again closed (${\nu}_{s}$ again coincides with ${\nu}_{0}$).
Beware: cycles are often called circuits [Cam94]; the distinction between circuits and cycles here follows Wilson [Wil02]. These closed wanderings are often called after their length: $s$circuits, $s$cycles.
The case $s=0$ is excluded from these definitions; $1$cycles are loops so imply a pseudograph; $2$cycles are double edges implying multigraphs; so $3$ is the minimum cycle length in a proper graph.
Note also that in trivalent aka cubic graphs^{} a closed trail is automatically a closed path: it is impossible to visit a vertex (in via edge $a$, out via edge $b$ say) and visit it again (in via $c$, out via $d$) without also revisiting an edge, because there are only three edges at each vertex.

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An open walk, open trek, open trail is one that isn’t closed.

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An open path (sometimes open chain) is just a path as defined above (because a closed path isn’t actually a path). Still, the term is useful when you want to emphasise the contrast with a closed path.
References
 1

Cam94
Peter J. Cameron,
Combinatorics: topics, techniques, algorithms
Camb. Univ. Pr. 1994, ISBN 0 521 45761 0,
http://www.maths.qmul.ac.uk/ pjc/comb/http://www.maths.qmul.ac.uk/ pjc/comb/ (solutions, errata &c.) 
Wil02
Robert A. Wilson,
Graphs, Colourings and the Fourcolour Theorem,
Oxford Univ. Pr. 2002, ISBN 0 19 851062 4 (pbk),
http://www.maths.qmul.ac.uk/ raw/graph.htmlhttp://www.maths.qmul.ac.uk/ raw/graph.html (errata &c.)
Title  (closed) walk / trek / trail / path 
Canonical name  closedWalkTrekTrailPath 
Date of creation  20130322 15:09:50 
Last modified on  20130322 15:09:50 
Owner  marijke (8873) 
Last modified by  marijke (8873) 
Numerical id  9 
Author  marijke (8873) 
Entry type  Definition 
Classification  msc 05C38 
Related topic  Path2 
Related topic  ConnectedGraph 
Related topic  KConnectedGraph 
Related topic  Diameter3 
Defines  walk 
Defines  trek 
Defines  trail 
Defines  path 
Defines  chain 
Defines  circuit 
Defines  cycle 
Defines  closed walk 
Defines  closed trek 
Defines  closed trail 
Defines  closed path 
Defines  closed chain 
Defines  open walk 
Defines  open trek 
Defines  open trail 
Defines  open path 
Defines  open chain 
Defines  $s$arc 
Defines  $s$cycle 
Defines  elementary cycle 
Defines  $s$circuit 