Two different animal species have a spectrum of inter-breeding possibilities:
1) Completely incompatible.
In many cases, it's logistically impossible for the two species to breed; try and imagine a giraffe and a dolphin, for example. Even if one could get the gametes (reproductive cells, e.g. egg and sperm in humans) together, no fertilization would take place.
Eggs have barriers around them containing specific proteins, and in order for a sperm to be able to fertilize the egg, the sperm must have matching enzymes that are capable of penetrating this barrier. You could think of it like a "lock and key." Only sperm of the same species have the correct "key" to enable their DNA to be transferred into the egg and allow fertilization to occur.
2) Fertilization occurs, but the zygote (fertilized egg) isn't viable.
This category covers a quite wide spectrum, but embraces all of the options between fertilization occurring but no subsequent cell division happening (ie remaining a single cell zygote), through to fertilization happening and a "nearly normal" but sterile adult resulting, as is the case with a liger. The factor that determines where on this spectrum the hybrid will fall is the degree of genetic similarity of the parents, particularly with regard to how the essential functional genes are allocated to chromosomes.
If two species have similar functions allocated similarly, to the same complement of chromosomes, you are more likely to end up with a viable adult. The greater the difference in number of chromosomes, and the distribution of essential gene functions on those chromosomes, the less likely that the zygote will be viable.
3) Fertile offspring.
If two species can create a hybrid that is fertile, then they're actually, by definition, not separate species.
Why is fertility so difficult for hybrids?
Fertility is the part of the process that requires the greatest degree of genetic similarity because of the differences between mitosis (asexual cell reproduction) and meiosis (sexual cell reproduction). Mitosis is the process by which a zygote turns into an adult animal by copying its own cells over and over and increasing in size; meiosis is the process by which that adult creates eggs or sperm to contribute to the next generation.
Chromosomes come in pairs; we have two copies of each chromosome, one from each parent. Humans, for example, have 46 chromosomes, consisting of 23 pairs. Both types of cell division, mitosis and meiosis, require chromosomes to line up with the analogue chromosome of their "pair" prior to cell division, then the pairs are "split down the middle," to ensure that each new cell ends up with complementary halves and a full chromosomal complement.
When you have chromosomes of two species, they will have some differences in how all the functions are distributed on chromosomes. If chromosomes are "similar enough" to be recognized as pairs by the cellular machinery, then for mitosis, the pairs can line up along the center of the cell that is about to divide, then duplicate, then separate into two identical groups to form two identical child cells. This might require, for example, for the two parents to have the same number of chromosomes, and the chromosomes to be of approximately similar sizes.
But the crucial distinction that makes fertility extremely challenging for hybrids is that during meiosis, material is exchanged between the two pairs of chromosomes. (Note that this diagram shows what happens to just one chromosome pair during these cell divisions, so in reality it's considerably more complex.)
As you can see from the diagram, during meiosis, parts of one chromosome pair are being swapped for "chunks" of its opposite number. When your chromosomes and genes precisely match up, that's not problematic - in fact, advantageous; that's how genetic diversity is created - but if the lion and tiger, for example, have a gene for an essential function in a very slightly different place on the chromosome, swapping a bit of one chromosome for a chunk of its opposite in the same physical location on the chromosome is almost certain to interfere with the function performed by that gene.
As every living organism has thousands of essential functions, the chances of interfering with at least one of them is very high unless the two species are incredibly genetically similar.
In fact, the most common scientific definition of a species takes advantage of this: if a hybrid of two organisms can undergo meiosis and produce viable offspring, then by definition, those two organisms are members of the same species. 
 The American Heritage Science Dictionary, definition of species: "A group of organisms having many characteristics in common and ranking below a genus. Organisms that reproduce sexually and belong to the same species interbreed and produce fertile offspring."