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 appeared, 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 can transfer copies of programs or their installers to any Mac we wanted too as long as it's not copy protected.
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 10.7 and above Apple installed a extra security feature.
You need to head to System Preferences > Security > General and set your Downloads (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!
(You can turn it back on afterwards or use the control click method on the installer.)
Good places for software choices are 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.
Another benefit is you get updates 4-8 weeks faster that AppStore as there is no wait, good for anything connected to the Internet especially.
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 or have legal recourse afterwards. Another tactic is some websites that pool free or developers software are adding their own wrapper to the installer to put adware/toolbars etc., onto your machine. Adobe does this with McAfee anti-virus installs on Windows PC's when one installs the free Flash browser plug-in, but not for OS X. OS X users in general don't like this sort of practice of something "else" being bundled with their original install.
Misguided places to download software is by trick buttons/links in advertising that is displayed on the site your intending to download software from.
One must be careful and watch the site, reload the page and watch where the advertising changes as not to accidentially click a download looking button in a advertisement or you will be downloading something you likely wont want.
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:
VLC free content and Icecast Internet Radio (more than iTunes!)
VirtualDJ free popular DJ software
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, CoconutBattery, 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.