My question is quite simple. I'm asking if I can use the power cable you get along with it as a USB power supply. Instead of plugging it to a power socket, can I use the USB to connect directly to my computer? I'm asking this because I'm not sure about the voltage and I'm afraid I might fry it accidentally.
The most general answer would be: it depends.
I personally would never rely on the regular USB port's ability to provide such level of power as the one required by the Model B, but Model A's much lower consumption would be under the usually known 500 mA limit on the USB port.
In my experience, a number of USB hosts that I have tried in the past aren't even able to supply the old nominal 500 mA of the standard. I feel this value changed from older to newer USB specifications, last one I read was version 2.0.
Consider that "check your power supply" is by far the most common answer I give to a myriad of different issues I am asked about concerning the Raspberry Pi. Just use a good reliable power source that can provide 1 A, if you don't want to run into random problems (from reboots to dying network adapters to SD writing errors).
USB, according to the spec, can only provide 500mA. But it depends on the motherboard you have in your PC. Most I've heard of happily provide more. If you draw more that the mobo is willing to give you, a fuse is tripped. Most mobos have polyfuses, so the fuse will reset after once everything is normal again. But there is a very small risk that you will permanently disable your usb port.
All USB ports work on 5V. That said, some ports may not be able to deliver enough current to the Raspberry.
I wouldn't rely on it for continuous operation. Worst case scenario the port's polyfuse on your motherboard will open the circuitry, and turn off the port for some amount of time. I doubt there would be any damage to the PC or the PI, but it depends on the motherboard (i.e. if it has said fuses incorporated into the design - some older, archaic ones may not have them). You'll probably be fine (at least for some time, since polyfuses have limited trip cycle, i.e. the number of times the fuse can be blown without a failure). In terms of doing things on Raspberry though - it's risky. Randomly turning off the device may corrupt the file system on your SD card.
Since laptop/PC USB ports are not suited to provide higher amperage, a better solution would be to use powered USB hub, or USBv3 high power port if you have one available on your PC.
If you draw too much current from USB port the voltage goes down by a significant amount, before the polyfuse kicks in. That's something to consider, since Raspberry PI's components doesn't work terribly well when on under-voltage.
Speaking of under-voltage...
Another thing to consider is the increased power consumption when the temperature goes up. The under-voltage scenario can lead to heating of internal components, both on Raspberry and PC - this leads to increased internal resistance, and even more power consumption. Due to this effect, the polyfuse openings can kick in randomly depending on temperature. The device may power on when cooled down, and then power off after temperature builds up. This may lead to random Raspberry shutdowns, and can be disastrous for SD card inside.
You can use (usually blue coloured) USB 3.x ports (at least those marked with a battery symbol) if your PC has one. They can provide more than the 500mA of a normal USB 2.x port.
I have an internal USB 3.x hub in my PC which suffices to power an Raspberry Pi 3 in most cases. (Occassionally under heavy load the symbol for not enough power showed up on the screen, but it never crashed or rebooted because of that.)
At least Thinkpads (maybe also other devices) also have yellow coloured USB ports (see https://superuser.com/questions/218053/are-usb-3-0-ports-yellow) which can also provide power if the laptop is not on. That's probably also to consider when choosing which USB socket to use to power an Raspberry Pi.