I want to expand my network and transmit it wired through my home and office (in another floor). I know the IT part kind of well, but don't know how to push the network cable through a conduit (already know where's the entry and exit). Do you need a special tool for this?
You don't push, you pull. If you have an assistant, they can push while you pull.
Rather than waste money on a fish-tape for a job with conduit in place, just connect a vacuum cleaner to one end and feed string from the other end. You can tie a bit of plastic bag to the end of the string to make it vacuum in better, but a bare string will work for short runs.
Check at the vacuum cleaner end and stop when the string is there. If you need more of a pull, use the string to pull in a rope - either way, attach to the cable and pull it through. Braided hollow poly rope is good, but if the pull is not unreasonable a strong string will work. Tie in and tape on well, keeping it smooth.
I have done this rather a lot. The only place I want fish tapes is if I'm trying to hook two of them in a wall without conduit that I can't just rip open for access. Pulling tape is more useful, but likely overkill for a job in your house (it's a flat, flexible tape somewhere between string and rope, often marked so you can measure distance with it, too.)
In most cases if a conduit has other wires in it, you will be better off pulling those out (use them to pull in a rope) adding the new cable, and pulling all of them in at once - adding a cable is considerably more difficult with other cables in place.
Depends on how bendy the path is, what's in the conduit already, and where any intermediate access points may be. In an empty conduit I can usually push wire around one 90. More than that gets difficult.
I have been known to pull and push the existing wires a couple inches, using the "push" motion to drag the new wires through inch by inch.
The first and most obvious tool is a "fishing tape", a stiff spring-steel wire with a hook on it. It's designed to be pushed through conduit containing existing wires. Then you hook and tape your wires to it, wrap this splice with enough electrical tape that it won't snag on anything, and pull the wires through. That's why they call this craft "pulling".
There is indeed a whole electrician's truck full of pulling tools - lube, "Chinese finger puzzle" wire grabbers, even power winches. The biggest mistake we often see from DIYers is using the smallest (cheapest) conduit that is legal for his/her wires, and maximum allowed bends... And then needing to call that truck because they've made the pull so difficult.
If it's a VERY short distance between the entry and exit, try a straightened-out length of coat hanger, some duct tape, and the end of your cable.
Fish tape is useful, as the others have stated. I wired a whole 3-bedroom apartment myself once, mostly from the attic. For some of the locations, I was running more than one cable (sometimes up to three) through drilled holes. Instead of using the fishtape multiple times (which becomes more difficult) through the same hole, I'd fish a length of nylon "parachute" cord through a hole, and then duct-tape my wires to the parachute cord at staggered interviews to keep the profile as flat as possible. Then I'd go back to the hole and PULL the parachute cord through.