Yes, a browser is a program that's used to go online and view web sites. Mac's come with Safari and some people prefer FireFox. Windows comes with Internet Explorer and some people prefer FireFox. There are many others available like Chrome from Google, and Opera.
So you are not downloading a game, you are trying to install it from a disc. Big difference in terminology. Since nobody can look over your shoulder we assumed you were trying to download a program over the internet.
The web site for your game is here:
http://www.diner-tycoon.com/
There is a "Contacts" link at the bottom of their page that you can use to ask them questions. The site says the program is good for 10.4 and 10.5 but doesn't mention 10.6 at all. That doesn't mean it won't work, but their site doesn't say so.
When you begin the install from the disc, you might be getting a warning that the installer requires an Admin account. It should also pop up another window with your User Account filled in and a place for you to type your Admin password.
Games like this could also require Rosetta (a technology Apple provides for compatibility with older programs or to run PPC code on an Intel Mac). If Rosetta is not installed on your Mac, another window should pop up asking you if you want to install it. Here again you'll have to type in your Admin password.
It's possible that the game is not compatible with Snow Leopard at all. Here is an earlier post about the same game, with no resolution:
http://discussions.apple.com/message.jspa?messageID=10394245
Hope some of this makes sense.