In the firefox developer tools, under the "Net" panel, resources that are loaded have their load time split into different colors/categories. These are:
What do each of these represent, and more specifically, does any of them accurately represent the amount of time that the server is thinking (accessing the database, running algorithms, etc)?
Thanks.
You couldn't accurately determine what the server is doing as such, I'm afraid.
You can discount most of them except Waiting, however, as the rest occur before and after the server handles your request. What it actually does while you wait will be a 'black box'.
There may be some asynchronous operations taking place during Sending and Receiving, so again it's hard to be accurate but you can get a ballpark figure of the time the server is working and the time the request spends travelling back and forth.
EDIT
Rough Definitions:
DNS Lookup: Translating the web address into a destination IP address by using a DNS server Connecting: Establishing a connection with the web server
Blocking: Previously known as 'queueing', this is explained in more detail here
Sending: Sending your HTTP Request to the server
Waiting: Waiting for a response from the server - this is where it's probably doing all the work
Receiving: Getting the HTTP response back from the server