Can Minecraft Pocket Edition play with a desktop Minecraft?

No, you can't. They're entirely different games.

To expand a bit on BlaXpirit's answer, the PC version, the Pocket Edition, and the XBox 360 Edition are all separate games, with separate features and in separate states of development.

The games are written in different languages for different platforms, and the XBox version is actually developed by a different company than Minecraft is, so it is (unfortunately) very unlikely that these versions will ever be able to join each other in online play.

There are now two different versions of Minecraft on PC

• The java version, the original one
• The windows 10 version, which can play along the Pocket Editions on Android and iOS

You cannot join their Java version on PC, but you surely can on the Windows 10 version. Of course, they need Windows 10 on their PC to install it

It is possible to get the Windows 10 version for free if you have already existing Minecraft accounts, just go on the website in account management and you should find the option

(There might be some elaborate way to play the java version on your ipad, but this would require so much work it isn't worth it, if you do get it working, you will still have the Controls issue (how will you control a game designed to work with keyboard and mouse?))