How many planets are in the universe? How is this number estimated?
We don't know.
First of all: all we can now about is our 13 billion lightyears neighborhood, since the Universe is 13 billion years old, and no information can travel faster, than the speed of light. This distance is called the horizon. We can only guess if there is finite or infinite space behind the horizon. If it's infinite, than most probably your answer is: there are infinte planets in the Universe. But this is something we can never know.
What we could know, if we would be smart enough, and what we are willing to know by finding out more about the Cosmos is this: how much planets are there within the horizon?
Since we know the volume of this region (it's a sphere with 13 billion lightyears radius) all we need to find out are these:
the density of galaxies (avarage number of galaxies in a unit volume)
the avarage number of stars per galaxies
the avarage number of planets per stars
If you multiply all of these you get your answer. Since all of these are still very uncertain, we can't say nothing sure yet, just these: there are so much planets, that are beyond our imagination.