Computer build
 

It works on my Amd machine.
Asus M4A89 GTD PRO/USB3

Well, I'd question an operating system being "good" based on the ability to run noddy online flash games. Linux is capable of a hell of a lot more than that, what with it being the most popular operating system used within enterprise these days. What you are asking for is very basic, most standard Linux desktop machines are perfectly capable of doing this. I'd suggest Ubuntu, it'll be fine.

Well, These aren't "Noddy games". These are MMORPG games. And here is the hardware I am planning to use on my computer (If I use Linux, the cost will be under $300):

Biostar A780L Motherboard

AMD ADX250OCGMBOX Athlon II X2 250 Dual Core Processor (CPU fan included)

Seagate ST3250318AS Barracuda 7200.12 Hard Drive (250 GB)

Patriot Signature 1024MB PC4200 DDR2 533MHz Memory (1GB)

I-Inc IK141ABB 14" Widescreen LED Monitor

Diablotek PHD 350W ATX Power Supply

TX3 FAN POWER SPLITTER CABLE (6 IN)

PowerUp G54-8019 Executive ATX Mid-Tower Case

2 Power Up 120mm Case Fans

If there are any suggestions to what I may need or, what I may consider to decrease to lower the cost, feel free to give your opinions.
And tell me specifically what version of linus would be well suited for me (preferably one that looks like a Mac).

The key things in your list are the hardware the OS actually uses.

Any modern Linux distro should have no problem with AMD, Seagate, or the monitor and doesn't care about the power supply etc. as long as the components are getting the correct amount of power and cooling.

The pieces of hardware that can be most iffy with Linux are wireless cards and printers. Some manufacturers are not very helpful when it comes to making drivers available, but that has improved drastically in the six years I've been using Linux. You can google "[hardware make/model linux]" and check the LQ hardware compatability list (HCL) for more information, but the LQ HCL is hardly definitive--it depends on volunteer submissions.

If you want to do some testing, you can download some Live CDs and boot to them from your optical drive; they will create a ramdisk in memory and boot to that. If they work, it will indicate that Linux will work with your set up. My first hands-on with Linux was a Knoppix Live CD. After one look, I knew I wanted me some of that and ended up with Slackware 10.0.

Never used a Mac, so I can't help with that.

Well, I want Linux version that looks like a Mac and can be compatible with an internet adapter, that can run all the requirements I said above. Please tell me a good distro for Linux that would suit me.

^this

All Linux versions are compatible with the Internet, all can be setup to look like a Mac, and assuming that all these games are Flash games, all Linux distros are able to run them.
Now it is really up to you to try a few of them, and see what fit your needs. Since you seem to be new to Linux I would at first recommend Ubuntu, Linux Mint and PCLinuxOS. I think that Ubuntu would be the most "Mac-like".
Just a side note, if you really want something that looks and acts like a Mac, I would recommend to buy a Mac.

Suggestion --
http://www.gaanza.com/blog/mac-ubuntu/

Stick with 10.04
&
double your RAM

In fact, SuperOS 10.04 may be better for you, yet!

OP-- ignore some of these and take what you need (newbies are fair game to some)

The hardware seems fine for anything: much faster than mine! If the games are on-line then you will be OK with motherboard video.

If you are concerned about the look of things (and that can be changed), you might like to download CDs of distros using the three major interfaces: the Gnome, Xfce, and KDE desktops. You can then try them from the CD, without installing, to decide which you like. For ease of use and variety, I'd suggest Ubuntu for Gnome, PCLinuxOS for KDE, and CrunchBang for Xfce. They all look and feel different, and I'm sure one will appeal.