Ping test

If you receive Request Timed Out, you could assume there is a problem with the destination device, whereas if you receive Host Unreachable, it could be a problem somewhere along the line before you reach the destination.

http://compnetworking.about.com/library/weekly/aa100700a.htm

"Request Timed Out"
Ping reports the percentage of packets acknowledged by the remote host. Typically this number will be 100% or 0%. When an Internet host pings at 0%, this does notnecessarily mean the server is "down" or unavailable. Internet Web servers especially may be configured to disregard ping requests for security purposes.

One scenario that will result in a ping response rate of other than 0% or 100% occurs when ahost is shutdown and leaves the network (or boots and joins the network).

It is also possible for ping packets to be lost in transit, causing ping to report a host as unavailable when in fact it is available but unreachable. One cause of lost or dropped packets is extreme levels of traffic; in general, ping utilities will be unusable on heavily-loaded networks. Another cause of lost packets -- the "too many hops" or zero-TTL condition....

And lots more info and examples at the above link

Request time out is when you ping an IP address that is on your own subnet which is either offline or doesn't exist.
Destination unreachable is when you ping an IP that is not on your subnet and you are going through a router/gateway to the required IP address and it can't be seen.

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