Wifi - or sometimes called WLAN (as in Wireless Local Area Network) is the part which would allow you to use a wireless router at home. 2G, 3G are concerning the facilities usable over the mobile (GSM) networks.
The 'using iPlayer at home' question concerning using Wi-Fi (or in this case, that the phone does not have it).
Unfortunately, the vast majority of phones do not include it, and I have yet to see many websites for UK stores include the use of it as a 'features' you can choose. For example, on Ebay, you can filter the range of phones to include those which have Bluetooth, or MP3 player, or Radio, how detailed the camera may be (from VGA to 1, 2, 3, 5 up to 12 Megapixels) but there's no similar tick-box to choose only models which include Wi-Fi.
Further, when a phone does include it, the cost seems quite a lot higher. You can get some touch-screen (but otherwise 'basic') phones for under 35 pounds but for ones including Wi-Fi, I suspect you will find few at under 100 to 200 pounds (on PAYG) or be tied into a costly contract, unfortunately.
(I have been looking for a PAYG phone with Wi-Fi support myself, but have seen nothing I can justify, just on [high] cost grounds, so will have to wait.)