An ADSL Router has a built in modem for telephone line broadband - YOU DO NOT WANT THIS.
The description of DSL or cable rouiter is often used for those that have an Ethernet WAN port to connect to an existing modem (what you want), but there are a few that can be found with a DOCSIS cablemedem built in (usable only where the cable operator will allow your own equipment on the cable)
The key thing in specifications - ETHERNET WAN PORT - 10/100 (/1000)
The Wireless specifications:
802.11b is an obsolete, 11 megabit standard, supported for legacy compatibility - on some setups, you can disable compatibility if not needed, for a small increase in thoughput (otherwise packets of faster protocols are framed with guard signalling to prevent legacy systems seeing the channel as clear.
802.11a was designed at a similar time as b, but uses a higher transmission band (5GHz instead of 2.4GHz) and allowed 54 megabit - higher speeds on 5GHz are now usually lumped in as "Dual band n)
802.11g brought 54 megabit to the 2.4GHz band
802.11n added further options, increasing basic througput agaian as well as allowing wider (double) channels and multiple streams - the two most common specifications being N-150 (wide channel, drops to 75 if you cannot use a wide channel) and N-300 (wide channel, 2 antenna, 2 spatial streams).
Each level is generally backward compatible - an N-router can support even 802.11b devices, though with some compromise to throughput on faster devices.