I wonder what are the different ways of defining the 3-manifold. Obviously for average human being it is difficult to imagine the 3-manifold. Therefore the presentation or visualisation of such object is important to understand what we are talking about.

Here are the possible ways which I know:

1) Dehn surgery on framed link in $S^3$.

2) Homeomorphism of the surface generate 3-manifold by gluing solid handlebodies along this homeomorphism - Heegaard splitting.

3) Gluing faces of polyhedron (is it somehow formalized ?).

4) Define real smooth function $f$ on $\mathbb R^4$, then $f^{-1}(p)$ is 3-manifold for not critical value $p$ (see Morse theory). More general try smooth function $\mathbb R^5 \to \mathbb R^2$.

5) Spherical 3-manifolds can be determined by finite subgroup of $SO(4)$.

6) Brieskorn manifolds looks as a nice way to define 3-manifold. Is there any generalization to obtain 3-manifolds by intersecting complex manifold of complex dimension 2 with hyperplane or sphere.

Such methods are preferred which could lead to step-by-step "manufacturing" of any 3-manifold. Methods 1) and 2) above satisfy this criteria. Method 1) is the best since we can visualize the link easily in dimension 3 and even 2 as link diagram with numbers.

Note: I have not mentioned hyperbolic manifolds, since I know very little about it.