Hi all, I have a small amount of arduino experience and I'm trying to find out what I'd need to research (and buy) for a little project I'm working on. In short I'd have an accelerometer take some measurements, feed into Arduino A, then have Arduino A wirelessly send that information to my computer . From there I'd like to send the information from A through the internet to another computer and wirelessly transmit data to another Arduino, Arduino B. If possible I'd like to utilize available wifi, but if that's not feasible I'd like to know what would be better (an xbee transmitter?). Thanks in advance for the help!
You might want to check out the RFXduino TCP/IP wireless gateway for Arduinos using nRF24L01+ radios at embeddedcoolness.com.
If you want to go through the Internet, you'll need something that speaks TCP/IP and/or UDP. If you want wireless, WiFi shields are the "traditional" (if rather poor bang-for-buck) approach, but now there are Linux based gateway solutions on offer as well, such as the Yun and RFXduino, offering greater power and flexibility.
I don't think I understood any of that haha. I have zero background in networking (and a very small background with Arduino) so if you could slow down and clarify/define a lot of the terms you use (TCP/IP and UDP) as well as what a lot of the alternatives mean (Linux based, etc) that would be really helpful.
I also just want to clarify my project that way I'm not too confusing about what I'm looking for. I'd ideally like to have an arduino wirelessly connnected to the internet via wifi sending information over the internet to another arduino wirelessly connected to a different network which receives the information.
So that being said could you help me understand what sort of hardware and software I'd need step by step?
In other words:
Step 1: How should I connect the arduino to the wifi (what hardware and software allow information to wirelessly be transmitted).
Step 2: How could I then transmit this information to someone else's network.
Step 3: And finally have this information be transmitted to an arduino via wifi.
This is the technical and most difficult part of this that you will probably not understand if you have no network experience.
Step 2: How could I then transmit this information to someone else's network.
On the destination (someone else's) router with the public ip assigned to it's WAN interface, you need to destination route the destination public ip and port to the private network ip and port of the target (server) Arduino. Then insure the destination router's firewall doesn't block that port from the internet.
Even if you know very little about networking now, you will learn a lot doing this. 
Step 1: How should I connect the arduino to the wifi (what hardware and software allow information to wirelessly be transmitted).
What is the "wifi" you are connecting to? Your wireless router or some other wireless access point? Where are wifi shields available for the arduino, or other ways using a combo of RF adapters or wireless routers.
Step 2: How could I then transmit this information to someone else's network.
Someone else's network will need to be setup to receive your information.
Step 3: And finally have this information be transmitted to an arduino via wifi.
Probably a reverse setup of Step 1.