I want to know whether I can get internet connection on my iPad Mini (WiFi only) through my router.
If yes, what are the criteria of that router?
Yes. If it is a wireless router, iPad Mini allows you to connect to any standard wifi.
As per Apple's technical specification page the iPad Mini can connect to a network that meets the following requirements:
802.11a/b/g/n Wi‑Fi (802.11n 2.4GHz and 5GHz)
For example, if you have a 802.11 2.4GHz b network, the iPad Mini should connect to it.