The number that is often cited to be finetuned 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 finetunedness 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.