For a map f \colon X \to Y, the mapping cylinder M_f is the quotient of the disjoint union of X \times [0, 1] and Y obtained by identifying each point (x, 1) with f(x) \in Y [1]. The end of the cylinder at t = 0 is a copy of X, the space Y sits inside M_f unchanged, and each point x of X is joined to its image f(x) by the segment from (x, 0) to (x, 1).
The identity map of X gives the cylinder X \times [0, 1] itself, and the constant map from X to a point gives the cone on X, in which all the segments meet at the vertex. For f \colon S^1 \to S^1, f(z) = z^2, every point of Y has two preimages, the two segments ending there continue each other, and M_f is the Möbius band, with X as its boundary circle and Y as its core circle.
The figure builds the mapping cylinder of z \mapsto z^3. Three segments end at each point of the core circle Y, and they open at angles of a third of a full turn in the plane perpendicular to the circle. Going once around Y, the three fins turn by a third of a full turn and are permuted, so their outer edges join into a single closed curve X that winds three times around Y.
The mapping cylinder contains both X and Y as closed subspaces, and when X and Y are CW complexes and f is cellular, M_f is a CW complex with both as subcomplexes [1]. A map that is not injective, such as z \mapsto z^3, becomes injective on the end X of M_f: points that f sends to the same point stay apart until the far ends of their segments.
Sliding each point of X \times [0, 1] along its segment toward Y by the maps (x, t) \mapsto (x, t + s(1 - t)), 0 \leq s \leq 1, while fixing Y, is a deformation retraction of M_f onto Y [1]. The final map r \colon M_f \to Y sends (x, t) to f(x), and it is a homotopy equivalence.
Let i \colon X \to M_f be the inclusion of the end. Then f = r \circ i with r a homotopy equivalence, so up to homotopy equivalence every map is an inclusion. Statements about maps can therefore often be reduced to statements about a space and a subspace, here the pair (M_f, X), which is how the mapping cylinder is used in homology and homotopy theory.
The figure shows the projection f of the circle X onto the segment Y = [-1, 1]. The segments from the points of X down to their images form two sheets that meet along Y, with a cross section shaped like a V. Two points of X with the same image (colored) are different points of M_f, and they meet only when the deformation retraction pushes them down to Y.
The deformation retractions in the article on homotopy are of this kind. The closed annulus is the mapping cylinder of the identity of the circle, the Möbius band is the mapping cylinder of z \mapsto z^2, and the solid torus S^1 \times D^2 is the mapping cylinder of the projection of its boundary torus onto the core circle. In each case the retraction slides points along the segments of the mapping cylinder.
The other end X is a deformation retract of M_f exactly when f is a homotopy equivalence [1]. Consequently two spaces are homotopy equivalent if and only if both are deformation retracts of a third space, and for a homotopy equivalence f \colon X \to Y this third space can be taken to be M_f.
Deformation retraction goes only one way, from a space onto a subspace. The constant map from S^1 to a point is not a homotopy equivalence, and its mapping cylinder is the cone on S^1, a disk. The disk deformation retracts onto its vertex but not onto its boundary circle, since a retraction of the disk onto the circle would make the identity of S^1 null-homotopic.
Collapsing the end X of M_f to a single point gives the mapping cone C_f = M_f/X, in which the cylinder becomes the cone on X attached to Y along f [1]. For a map f \colon S^{n-1} \to Y the cone is the disk D^n, and C_f is Y with an n-cell attached by f. For the constant map to a point, C_f is the suspension of X, and for z \mapsto z^2 it is \mathbb{RP}^2.
The figure shows the inclusion f of the equator into S^2. Its mapping cylinder is the sphere with a ring attached along the equator, and the outer edge of the ring is the end X. Shrinking that edge to a point turns the ring into a disk spanning the equator, the cone on X. The disk is contractible, and collapsing it pinches the sphere at the equator, so C_f is homotopy equivalent to S^2 \vee S^2.
When f is the inclusion of a subcomplex A of a CW complex X, the mapping cone is homotopy equivalent to the quotient X/A [1]. This is one reason quotients by subcomplexes appear so often in homology, in the long exact sequence of a pair and in computations for spaces built by attaching cells.