How would I be able to get data from another website and put it on my own website? Website A has profiles of users along with other stuff. I want to be able to get the data of the user and incorporate it in my site.
To get data from another website, you can use a RESTful approach by using the same protocols a human user would use in order for two machines to communicate.
The beautiful thing about URLs is that machines can use them too!
Since you've used the PHP tag, I'm assuming PHP is your server-side language of choice. The PHP cURL function will allow your server-side portion of your website to make HTTP GET and POST requests to a specific URL to get the data that URL represents. Once you have that data, you can then parse that data, store it, display it, or do whatever you want to do with that data.
This involves programming, and also depends on the specific type of data that you're trying to obtain; therefore, please review the cURL reference for more detailed examples that will help get you started on solving this problem.
Once you have more specific questions, with code examples, please post them on Stack Overflow.