Simplicial sets are an extension to simplicial complexes.
Model Category
A model category is a context in which we can do homotopy theory or some generalization thereof; two model categories are ‘the same’ for this purpose if they are Quillen equivalent. (nlab) 
A model category (sometimes called a Quillen model category) is a context for doing homotopy theory.
It is a category equipped with three classes of morphisms, each closed under composition:
 weak equivalences
 fibrations
 cofibrations
Quillen Equivalence
We need a mapping that preserves the structure of a topological space: the cycles, the boundaries, the holes and so on. However it does not need to conserve the exact points, faces of the complexes that hold them. The two shapes on the right are the same from a topology point of view they are a disk with a boundary round it. 

In order to do this we map the connections between the dimensions in the chain (boundaries), not the dimensions (points, faces, etc.) themselves. 
Simplicial Set
X: Δ^{op} > Set is a simpicial set
where:
 Δ is a finite totally ordered set {0,1,2...,n}
See page about chain complexes.
Chain Homotopy
Weak Equivalences
Usually equivalences are defined in terms of two functors in opposite directions however a weaker notion of equivalence is defined in terms of a functor going in one direction only.
Fibrations
A Serre fibration arises when we reverse the functor defining the weak equivalence.
Cofibrations
In any model category:
 A fibration that is also a weak equivalence is called a trivial (or acyclic) fibration.
 A cofibration that is also a weak equivalence is called a trivial (or acyclic) cofibration.
Cubical Type Theory
Cubical type theory extends these concepts with the idea of an interval [0,1]. This is discussed further on the page here.