What are the different ways of defining 3-manifolds? 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.
 A: This question ought to be community-wiki, since there is no correct answer. 
As indicated in the comments, one may represent a 3-manifold as a triangulation (simplicial complex), special spine, branched cover, real algebraic variety,  mechanical linkage,.... In fact, Thurston showed the existence of ``universal links", such that any 3-manifold is a branched cover over $S^3$ with branch locus the link (such as the figure 8 knot). It is known that any closed 3-manifold is a 3-fold branched cover over $S^3$. Also, the Borromean rings with orbifold locus of order 4 along each component were shown to be a universal orbifold. 
There are various combinatorial encodings of triangulations called gems and crystallizations.
Turaev introduced a notion of shadows to describe 3-manifolds. For relations with other presentations of 3-manifolds, see the paper of Costantino and Thurston.
A quite powerful way related to 5) to represent closed 3-manifolds is via the geometrization theorem. One considers orientable 3-manifolds (with torus
boundary) admitting geometric structures (on the interiors), and then
glue together by homemorphisms of tori (or to Klein bottles), and then connect sums. 
 Although it is somewhat trickier to describe geometric 3-manifolds this way, 
the advantage is that one may reproduce 3-manifolds without duplication (if done with care),
unlike the other techniques. 
2) is called a Heegaard splitting. 
For 3), there are notions of polyhedral representations 
of 3-manifolds (even allowing self-gluing of faces). 
For example, Cannon, Floyd and Parry showed that any
3-manifold admits a "bi-twist" description. 
Not every 3-manifold may be obtained by 4). 
6) is an interesting question: links of singularities are
restrictive, but I'm not sure what's known about intersecting
with a general hyperplane or sphere. 
