If you have DSL or a Cable connection, then there might be a router built into the modem. Otherwise you probably would know if you had a router or not, as it is a piece of hardware that allows multiple computers to connect to the internet using a single connection.
The timed out error, means that your computer didn't get a response from the server in a timely manner.
The unreachable error, means that your computer got the response from the server, but the server wasn't able to relay network information. This sometimes happens when a server on a network drops (ie: Net split).
Please note that these aren't the only possibilities, but they are the most common.