Instalation:

Make sure you have Java installed (if you can play Minecraft you have Java installed already)
Download the file, preferably with the download button, but if you have problems use the mirror
Open the compressed file (.zip). Your computer should be able to do this without external programs, but 7-zip and WinRAR can handle it as well.
Put the files anywhere on your computer, but note that it will be a lot easier to get 3D preview working if you put it in a folder you have write access in (For example, under Windows 'My Document' or 'Desktop' works. You can put it in a sub-folder. Under Linux any folder under your home folder should work. Just make sure to get all files if you want to get the most out of SkinEdit (such as the 'backgrounds' folder for additional backgrounds)
Just doubleclick 'MCSkinEdit.jar' (Might be displayed as 'MCSkinEdit'). SkinEdit should now launch
To get 3D preview working, just click ok when SkinEdit prompts you to enter the URL for the previewer. This is only done the first time you launch SkinEdit.