The first four of these models were discovered by E. Beltrami, although only the fourth one is traditionally identified by his name.
The first three models are more elementary from the point of view of axiomatic geometry. The other two live in modern differential geometry.
The pseudosphere provides us with a tangible trumpet-shaped surface in Euclidean 3-space that locally has a geometry of constant negative curvature. Unfortunately, only a small part of the full hyperbolic space can be embedded in this way, so the straight lines cannot be "indefinitely extended" and hence the global hyperbolic parallel property is not really demonstrated. The hyperboloid is a genuine, complete realization of the full hyperbolic geometry, but the 3-dimensional space in which it lives has an indefinite metric, like the space-time of special relativity. [punt]
From the point of view of differential geometry, there is a close symmetry between elliptic and hyperbolic geometry. More precisely, we are dealing with Riemannian geometry, in which lengths are defined by an arc-length formula to be integrated along a curve. The metric also makes it possible to define geodesics as curves that are the closest possible analogues of straight lines in a curved space. In a hyperbolic (negative curvature) space, geodesic lines that initially look parallel eventually veer away from each other (cf. alHaytham-Clavius, above). In an elliptic (positive curvature) space, geodesic lines that initially look parallel veer toward each other and eventually cross (as do meridians on the earth). It follows that in the hyperbolic space, triangles have angle sums less than 180 degrees (and are increasingly skinny as their dimensions grow), while in the elliptic space, triangles have angle sums greater than 180 degrees. (Consider a triangle with one vertex at the north pole and the other two on the equator.)
This nice picture deteriorates in axiomatic geometry ("Euclidean" geometry as carefully reaxiomatized by David Hilbert at the end of the 19th century, building on important work by Moritz Pasch (1843-1930)). The axioms imply that two lines intersect in at most one point, which is violated by the great circles on a sphere. (This problem can be eliminated by replacing the sphere by the projective plane, but then there's a problem with another axiom that requires each line to divide the space into two separate halves.) See Greenberg's book for details. This is the reason why Saccheri (Sec. 18.1) was able to prove existence of parallels but not uniqueness.
One more technical point: Somewhere Stillwell says that the parallel postulate is equivalent to "Angle sum of any triangle = 180 degrees." In fact, all that can be proved from the elementary Hilbert axioms is that "Euclid 5" implies the angle sum formula, not the converse. The problem is that the elementary axioms uniquely characterize the standard Euclidean and hyperbolic planes (depending upon whether Euclid 5 is affirmed or denied, respectively) only when supplemented by a more technical "continuity axiom" stating that every line segment has the topology of the real numbers (as established by Dedekind in the 19th century). If that assumption is removed, other models of the axioms become possible, notably the Dehn models (after Max Dehn (1878-1952), a student of Hilbert who later became a famous topologist). In these spaces parallel lines are nonunique (the usual definition of "hyperbolic") but nevertheless the angle sums of triangles can be always equal to two right angles, or always greater, as well as always less. Roughly speaking, these models are created by replacing the real numbers by a non-Archimedean field containing "infinitesimal elements" and then restricting attention to an infinitesimal neighborhood of a point. [time to punt again]
An ironic consequence of the foregoing is that the parallel postulate can't be proved within the Hilbert axioms, unless Euclidean geometry itself is inconsistent. In other words, if Saccheri et al. had succeeded in "vindicating" Euclid by proving his fifth postulate, they would have destroyed Euclidean geometry by proving it inconsistent.
Note that in these models the primitive terms (especially "straight line" and "congruent") are reinterpreted to mean something rather different than they mean in the embodying Euclidean spaces. This may leave the impression that the "true" geometry of a plane really is Euclidean, and hyperbolic geometry describes something different, so why the big deal? If you define Euclidean geometry as the geometry of the vector space R2 with its standard inner product, no one can argue with you.
On the other hand, now that we know that the parallel postulate is not inevitable, or built into the framework of the human mind so that denying it is inconceivable, as so many mathematicians and philosophers used to say, whether the geometry of physical space is Euclidean becomes an experimental question. Logically, it is entirely possible that space is hyperbolic, this time with the primitives meaning (almost) exactly what they always have in Euclidean thinking. In fact, in modern physical theory (general relativity, cosmology) space is something even more general, not homogeneous (i.e., may be different in different regions) and possibly changing in time. It can be "bumpy" as well as curved. In other words, nature is not exactly described by the R3 geometry given by linear algebra. On the small scale, planets etc. create bumps in the geometry, and the influence of the bumps on moving bodies constitutes the gravitational force of the planets. Gravity = nonhomogeneous geometry. On the large scale, however, current observational evidence indicates that the average curvature of the observable universe is very close to flat; but there is no good theoretical reason why that must be true, and elliptic (usually spherical) and hyperbolic 3-spaces are frequently studied by serious physicists. (Because the universe is expanding, the geometry of four-dimensional space-time is not flat.)