To "purchase" something means to buy it. The word "purchase" is more formal than "buy". Use it when you're communicating with a customer about something that they bought:

Which item did you purchase?

As a customer, you use "purchase" when you're making a formal complaint to a company:

I purchased one of your two-person backpacking tents through your online store last week, but when it arrived there was a large tear in it.

Another place where you'll often see the word "purchase" is in written descriptions of things that people or companies bought, like in newspapers:

Apple has purchased Quattro Wireless for $275 million dollars.

You can also use the word "purchase" as a noun, without any changes in spelling or pronunciation. For example:

Thank you for your purchase.