The number that is often cited to be fine-tuned is the ratio of the fine structure constant, \alpha_{EM} and the analogous constant for gravity, \alpha_G.
What are these constants?
In short, these numbers parameterize how strong particles interact with a particular field. i.e. \alpha_{EM} parameterize how strong a particle interacts with the electromagnetic field, while \alpha_G parameterize how strong a particle interacts with the gravitational field.
There are also analogous numbers for the strong and weak nuclear forces. Here is a list of their numerical values.
These numbers have a few important properties:
- They do not depend on the particular system under consideration (.e.g not based on the relative strength of gravitational or electromagnetic forces at some distance away from some particular particles)
- Their values cannot be predicted by the Standard Model, and instead must be experimentally obtained
- \alpha_{EM} is much bigger than \alpha_G by a factor of ~10^{36}; this simply states that electromagnetism is much stronger than gravity
Why are they fine tuned?
The idea is that if the ratio \alpha_{EM}/\alpha_G is even a little off its current value, the Universe will be very different than our Universe. Indeed, such alternative Universe would probably not support life. See e.g. the graph in @pevaquark’s post.
Can we explain this away?
The most popular explanation is that: during inflation an extremely large number (which could be infinite) of Universes pop into existence, each with possibly different values of \alpha_{EM}/\alpha_G. Because the number of Universes is very large, even if the chances are small certainly there are some Universes with the proper value of \alpha_{EM}/\alpha_G for life to exist. Because we must exist in these Universes, it is certainly not a surprise that we live in a Universe where \alpha_{EM}/\alpha_G is such that life can exist. This is an example of an anthropic principle.
Criticisms
The most common criticism is: we have no strong evidence for the existence of these other Universes. This is a valid criticism. However, the idea is that the current understanding of cosmology is consistent with explaining away the fine-tunedness of the Universe.
Further note
We do not even know what is the probability for a Universe to have a certain value of \alpha_{EM}/\alpha_G, so in a sense, all of this argument is moot.