The Type R and Type S badges are specifically Honda / Acura monikers.
The Type S generally speaking means that Honda added some extra horsepower to the car and some other various minor changes to make the car feel more sporty. Its similar to a "Performance Package" from BMW although not exactly the same.
Examples:
Acura CL / TL Type S - Came with 35 more horsepower (260hp/232 lb-ft) than the "Premium" model, "sport" seats with perforated leather, 17" rims, a sportier suspension and some other small things. (2001-2003 for the CL Type S; 2002-2003 for the TL Type S)
Acura RSX Type S - Comes with 40-50 extra horsepower (160hp versus 200hp in 2002-2004; 160hp versus 210hp in the 2005 models), 6 speed manual transmission standard in Type S (no automatic available), subtle bodykit standard on Type S models, in 2005 the RSX Type S gets 17" rims - there's some other various smaller changes that I can't remember
There have been some other Type S models available around the world, including an NSX Type S (Was only available in the USA as the NSX Alex Zanardi Signature Edition in 1999), and a Prelude Type S.
The Type R models are generally more extreme cars aimed at being put onto the track. Many Type R models do not have any additional horsepower, rather they are lightened to improve handling and performance many times at the cost of comfort. For example, sound deadening material in a Type R model might be stripped out to shed weight at the cost of increased noise while driving on the highway. Stiffening the ride through the use of stiffer shocks and increase spring rates are another way that the Type R models may improve handling while decreasing road comfort (think of how it feels riding a Lexus over a bumpy road versus riding in a lower end car like a Civic or Corrolla).
As others have said, only the Integra Type R has made it stateside, and as far as I can recall only Acuras have so far received the Type S moniker with most new Acuras now receiving A-Spec packages as the "sport" package, which is really just some cosmetic changes (like a bodykit), summer tires on bigger rims, and some lowering springs. Honda has the Honda Factory Performance package which is similar to the A-Spec package as well.
With the new Honda CEO saying that he is concerned with Honda becoming too risk averse like Toyota and saying he expects to surprise the automotive world with new projects, I'd expect Honda to go back to Type S and Type R models or something similar.
Hope this helps ... If you have anymore questions you can just do a search and come up with a lot of information on various Honda Type S and Type R models.