When you delete a file, the information is not immediately removed from the disk. Instead, the OS/file system simply updates a database keeping track of files on the disk to acknowledge that the file is no longer needed and hides the file from being visible. The information is only removed when, at some point in the future, the OS decides to use the space to store another file. That could be a few minutes later, or many weeks later, depending on how the computer is used. Before then, the data is still recoverable using data recovery programs.
When you do a wipe (also sometimes called a secure delete), you are telling the operating system to not only update its file records, but also immediately overwrite the disk space with either zeros or random data, making it much harder to recover anything.
To do a wipe many operating systems support the key combination Shift+Del which will immediately overwrite the space the file occupied. For maximum security though, there are many programs out there that actually fill the entire hard drive with random data and then delete the random data, sometimes more than a dozen times. This makes it nearly impossible to recover any previously deleted files. CCleaner, for example, is one freeware program that has that ability.
So to answer your followup questions:
1. Yes, it is usually quite easy to recover deleted (not wiped) data as long as it hasn't been too long since the data was deleted. This is how most data recovery programs work.
2. That depends. If you do a single wipe (ie, you overwrite the space once with random data), you're probably not going to be able to recover anything with standard data recovery programs anymore. However, skilled investigators may still be able to recover the data by putting your hard disk platters under an electron microscope to determine what was originally there, as overwrites are not always completely clean. It's like erasing a letter with a pencil and writing another letter over it; sometimes you can still faintly see the original letter.
If you do multiple passes with a very secure wiping algorithm though (like DoD, which overwrites the data 35(!) times), even this becomes unfeasible.