Install Bin Files in Linux

There are two types of bin files, self extracting archives and programs you run as they are, I'll mention both...

Steps

If the bin file is an installer/self-extracting archive, first download the thing and keep it somewhere safe just to avoid having to download it again.
Enter the terminal.
Enter Root mode, like so : su - (the hyphen is required) and provide the root password.
If needed copy the bin file to its final output folder - packages like the Java Runtime Environment require this. Read the online instructions first...
Change directories (folders) to the one containing the bin file, like so : cd /topmost/folder, for example cd /usr/share
Give the bin file execute permissions: chmod +x thefile.bin
Execute it: ./thefile.bin - the dot-slash has to be there
If the bin file is the program itself, chances are the file is compressed, untar/unzip in the destination folder, Firefox comes like that.
Copy the archive and unpack it in the output folder, that should produce a folder.
Enter the folder, locate the program - it is a bin file, give it execute permissions if need be (see step 6).
Make a starter for convenience, right-click on the desktop, select the option you need and follow the lead - an icon should appear.

Be careful where you put things - an unzip could overwrite stuff you need
If the program needs to be run system-wide, put it in a central spot , /usr/share is a good candidate
If you're a system-admin, avoid that the users do this...it'll mess up the system
Use this as a last resort, try to stick to the repository of your distribution of linux if at all possible.