Having had a quick look at the specs on-line - it appears the Vita has wi-fi enabled. In which case - assuming your computer also has wi-fi - just turn both systems on and you should be able to transmit the data that way.
Without a too-technical explanation, your router is what you use to connect to the internet. It is plugged into the phone line, and is either connected to your computer by a cable (ethernet) or issues a wireless signal that a wireless card in your PC connects to. So it is the same for your PSVita, you want to connect to that same router using the same SSID, password and other credentials.
you need a wifi connection and then get on internet from psp
You can't run unsigned code on a PSVita, that means you need to download games from the PSN.
Yes, you can get them from the PSN.
A PSVita does not have a router, a router is something you usually get from your internet service provider.
The PSVita connects to your wireless router in exactly the same way as your PC and other online devices, and uses your own internet connection to access its online functions such as the browser, PSN store, and online gaming. There is no additional cost for this.
The PSVita is not region locked, a US PSVita can play an EU PSVita game as would be bought in the UK.
Yes, the PSVita has physical games, small cartridges called PSVita Game Cards. The PSVita has a slot at the top for them.
The 'internet' is global system of interconnected computer networks. Home computers connect to the internet using a router. If you have a wireless router, the PS Vita has its own wi-fi capabilities, and can connect to the internet via your router. There is also a 3g version which can connect via the 3g network, but that needs a sim card and a subscription, like a mobile phone.
No, at this moment in time, is it not available for PSVita.
Yes, the PSVita's screen is HD.
Remote Play is where the PSVita is paired to a PS3 or PS4. The main console plays a game, and the PSVita acts as both controller and screen.