What is the equivalent of an "exe file"?

I would summarize the system in the following way:

Linux extension    |Windows Equivalent    |Short description
------------------------------------------------------------
.so, .o            | .dll                 | Object that can be loaded at runtime (Similar to DLL)
.a                 | .lib                 | Static library
[none], .elf(rare),| .exe, .com(rare)     | Linux executables
  .bin(rare)       |                      |
.sh                | .bat                 | Shell script
.exe               | .exe                 | Mono application, Wine application
.deb               | .msi                 | Installer package for Debian/Ubuntu releases
                   |                      |   (Though .deb is much more powerful with 
                   |                      |   native support for dependencies and repos). Note that .deb is actually a .ar archive with a special control file, a special file order, and a different extension.
.rpm               | .msi                 | Installer package for RedHat/CentOS releases.
.tar.gz, .tar, .gz | .zip                 | Compressed files that can contain a program or any other data, like images, documents, etc
.ko                | .sys                 | Drivers and kernel modules are loaded into the Linux kernel and have more hardware access than other programs.
.sh, .php, .py, etc| .bat, .cmd, .vbs,    | Linux is capable of running any file that it has an interpreter for. 
                   |   .js                |   A line at the top of the file called the shebang specifies what interpreter to run the file with. 
                   |                      |   Windows only supports .bat and .cmd files as Batch files, and .vbs (vbscript) and .js (JScript, not to be confused with JavaScript) via the Windows Script Host (WSH).

Any file that starts with #!(called a shebang or hashbang) on the first line is just a script. The path after the shebang is a path to the program (executable) that needs to parse the script, assuming that the execute permission has been set on the file with chmod a+x filename.

Therefore, answer your original question, there is no extension, or rarely, it is .elf or .bin.

This has been made Community Wiki, so anyone can add an entry.

There is no standard File-Extention like an ".exe" file in Windows.

On Linux nearly any file can be executable. The file ending just describes (but not necessarily) what or how a file is "executed".

For example a shell script ends with .sh and is "executed" via the bash shell.

In your question you ask for .deb and .tar.gz Well, the .deb file installs software on your system (Please be careful!) And the .tar.gz file is a compressed file like a .zip that you could know from Windows.

The concept of an executable is different in unix/linux than Windows.

Windows

Anything that ends in .exe or .com becomes an executable file.

Linux/Unix

Each and every file has an executable bit, so any file can be executed, unlike Windows. To see if a file is executable, you can check its properties (Permissions tab), or even see them marked in the terminal (they are all marked with a *).

Even text files (like shell scripts) can have their executable bits set, and be run as one.