Eurail is not valid in busses, with the exception of busses run by the railways themselves but you won't find those in cities.

Eurail is valid in trains of participating companies, including their trains within cities.

This differs quite a lot per country. But in Roma the urban buses/metro are so cheap, that you are penny-scratching. But f.e. in DE=Germany nearly all bigger cities also have a S-bahn-mostly run by DB and these may fine tune or run like a metro into the centre of town and other suburban areas. These are included, but for just a short local trip a normal ticket would be far more cheap as using a passday. And DO note that besides these trains here wil also be local trams/metro or whatever, not included.

ANY major and 99% of more regional cities here in EUR have an extensive local bus/tram/metro system that will cover all of town and sets its own fares. The London and Paris metro are famous all over the world-in P is quite cheap, in L astronomically expensive and even has no tickets anymore-you simply use your bank/creditcard to pay for the rides, by checking IN and OUT at gates

What about using the Eurail inside the city on the day you arrive/leave a city (so you're not using extra travel days)? I was going to ask the original question, but for Madrid, Paris, and Amsterdam. I know the cities have metro systems that I can buy tickets for, but if I need to travel on the metro in that city on the same day I arrive/leave that city, will I be able to use the Eurail pass? 

On a travel day, you can use Eurail in as many trains as you want, but only in trains of the participating companies. Metro systems however are not run by the railways and can therefore not be used with Eurail.

Ah. Thank you for clarifying!