Hello and welcome to my User Tip
I will explain how to download and install programs from the world wide web on your Mac.
Before the AppStore, we all downloaded software via the Internet web sites, that still is the case today. We also have to verify and trust the source of those downloads. We also could transfer copies of programs to any Mac we wanted too, provided the developer didn't install copy protection or involved installs all over the machine in various places.
A word of caution. Once you give your Administration Password to a any installer or software, it can do anything it likes to your machine's software and even firmware. So all you need to do is verify the source as a reliable place to download software from and I'll give some great sites with thousands of choices of verified software.
Some software installers require your Admin password, some it's a self contain program you can drag and drop into your Applications folder, which is a global access folder for all user accounts on the machine.
On Mac's with 10.8 Mountain Lion, Apple installed a extra security feature.
You need to head to System Preferences > Security and set your Gatekeeper to "Anywhere" as at this point your going to trust the source of your own downloads, and there is plenty of great places to get trusted software online!
The object of Gatekeepers other settings (AppStore only and AppStore and Identified Developers) is to allow unsure/new computers users to initially trust Apple for their source of software, sort of like training wheels so new users don't get into trouble.
There is a whole vast world of tens of thousands of available software choices in Mac software from other trusted and verified sources that existed long before the rather recent Gatekeeper and AppStore came along.
Some software you may want or need will never be or as fully featured on the AppStore; or the developer never will be or want to be, granted the favorable "Identified" status by Apple, however that doesn't mean the software the developer carries is malicious.
There are many, many reasons why a lot of some types of legitimate software will never be on the AppStore, or why a developer would rather deal directly with it's users and vise versa than having to go through Apple for everything.
Some software is just too complicated or involved for one. It might be a trial version so you can test it before you buy for instance. Another reason be it competes with Apple's offerings, or Apple feels it's not in the interest of all users or conflicts with partners interests that Apple deals with.
Who knows? As long as it's not malicious then your right to use it is your choice and not anyone else's because it's your machine and your the boss of what runs on it.
Apple knows this, but they create computers so young folks are safe, once you realize this and seasoned enough to judge where to install software from, it's there for the taking. One can always turn Gatekeeper back on afterwards anyway.
Most developers are out to make a profit, or trying to do good for others and thus have a forward and up front agenda, been around for some time. They usually use a central review and download site where users go to find multiple possible software solutions to their problem.
Good places for software choices are CNET Downloads, MacUpdate.com, Softpedia.com and Sourceforge.net, even visiting the developers site after that directly. The advantage there is you get more honest opinions, get to deal with the developer directly for troubleshooting or license issues, and in most cases don't have to deal with problematic copy protection issues that all AppStore apps are required to have.
Bad places to download software is by visiting warz sites, installing video players/updates on adult sites, P2P networks, links in people's posts on untrusted sites or any other method that you can't verify the source of the download.
CNET Downloads, MacUpdate.com, Softpedia.com are commercial sites that provide user feedback areas, verification etc., so check those reviews out before downloading to your machine.
So first you need to visit the site, find the software you want and decide to click the link to download it.
Make sure the version is compatible with the operating system version running on your machine and any hardware requirements.
Once it's downloaded, it will be inside your Download folder next to the Trash can, double click to open it,
Then double click on the file inside if it's a installer package or a compressed file, it should uncompressed or get the free Unarchiver to uncompress it.
Next it sometimes will pop up a window to say to drag to the Applications folder, or the installer program will run and you do that.
OS X will ask if you approved of this download, and it will check to see if it's part of it's database of malware and cancel it if it is.
Follow the direction of the installer if it needs your Admin password or not.
Now when your finished installing, click and drag the installer package desktop icon to the Trash can, this turns into a Eject so it ejects the .dmg volume.
Next in your Downloads folder you have to decide if your going to keep the installer package or Trash it, or move it to one of your other folders like Documents
You need to keep the Download folder clear and empty and not store anything there for long, because bad or hijacked web sites can cause a download without your knowledge and on high speed connections you won't even see it sometimes it's so fast.
If a Trojan is lying in wait for you in the Downloads folder, then you go to to double click on it to see what it is, it could compromise your machine, so keep the Downloads folder clean and clear except when in use.
Some excellent free software to test your web site downloading skills on:
GIMP image editor for OS X, free
LibreOffice (free office like suite, works with Office files)
Audacity - free audio program
Firefox (highly customizable free web browser with the most add-ons, themes and personas)
ClamXAv (free antivirus)
VirtualBox (free virtual machine software to run Windows or Linux in a window on OS X)
Cinebench (free Mac performance testing software)
Cube 2 Sauerbraten (free 3D game, runs fast even on older or less powerful Mac's)
Nexiuz Classic (another free 3D game, also runs fast on older/slower system)
Search MacUpdate.com for the free TinkerTool, OnyX, Hardware Monitor, , Grand Perspective, VLC, XRG, Easy Find and Caffeine.
Just the tip of the iceberg, but the most frequently used programs a lot of people have a need for.
Enjoy and Good Luck.