Type of Router required for converting Wired LAN to Wi-fi.

HI, when you are talking about home internet devices these terms can get a little bit mixed up.

It depends what your current internet connection is.
Please note i'm based in the UK so my knowledge is based on what's common here.
I am also assuming you already have a router which makes the connection for you as opposed to an USB modem.

If the internet connection on the wall is a telephone socket, possibly shared witha telephone using a microfilter/splitter
http://www.cablesdirect.co.uk/Write/Images/BT-2wire.jpg
http://www.idnet.net/support/guides/images/splitter.jpg

then you have ADSL. If you buy an adsl router with built in wifi, this will do the job. Please note the following standards and the max speed you will get from them (these are from memory so could be wrong):

ADSL = up to 2mb
ADSL 2 = up to 8mb
ADSL 2+ = up to 24mb

You need to buy a router which is capable of going up to whatever speed you are paying for, though most routers now are ADSL 2+ and backwards compatible.

Alternatively if your internet connection is an ethernet (RJ45) socket on the wall, or uses a cable modem which has an Ethernet socket on it. You need a router which is usually labeled as a cable modem again with built in WI-FI, its quite obvious when you see these, as they will have an ethernet socket, which is a different colour from the rest on the back instead of an RJ11 or similar. These are the most common types or router when buying online.

A wireless access point is another way to go, this will have an ethernet socket that connects to your existing router, anyone who connects to the access point will be on the router network and it will work just as well if not better than the all in one solution, just it may require additional setup.

Please be aware if you are considering buying your own router you may need to know the username and password for your connection which your ISP may not give you. These are often generic and some quick googling should tell you what they are or how to get them for your ISP.

if buying anything id get something that has wireless N as it is faster than wireless G, there is a new standard, wireless "AC" which is very fast but is not yet common in laptops etc.
I recommend the TP-Link brand for value for money.