Can I store and run applications on an external drive?

I am thinking of getting a this MacBook but the hard drive space is not sufficient. I was wondering if I bought an external hard drive for it would i be able to download apps and things onto it.

You can store and run applications in the non-standard /Application location on OS X, which means you can keep things on your external drive and run them from there.

You just have to be aware of the limitations of doing it this way:

1. The applications and data on the drive won't be available when it's not connected.
   
   Seems obvious but this can come up in surprising ways so I think it's worth saying here. Let's say, for example, that you have a text editor installed on your external drive and it's the default application for text files. When you double click on a text file when the drive is not connected you'll get an error about the application not being found. You'll need to right click and select an alternate application in that case.
2. The bandwidth, especially the sustained rates, for accessing the data and applications on this drive is limited to the connection method.
   
   For Thunderbolt it's pretty darn good but still not as good as a drive in the machine. For USB it's hit and miss. It depends a lot on the USB chipset used in the drive and what else is sharing the USB buss. Most applications will load resident in to memory when you launch them so for applications on an external USB drive you'll have some start up latency and that's it. But if you have to access a lot data or if the application loads itself in pieces as you access different features, you might find the I/O wait tedious. Expect beachballs here and there for sure.

Most applications can be safely relocated to the external drive by dragging them from /Applications to the new location on the remote storage. Applications are bundled in a nicely self-contained way. Note though that relocation the applications .app bundle from /Applications to the remote drive only moves the bundle. The configuration and ancillary data the application uses, generally found in /Library or ~/Library will remain on your primary drive. That data is harder to move. It's also usually smaller than the application bundle.

Yes - OS X doesn't care where you store apps for them to run.

Similarly, most programs don't force you to store all documents in one place.

You shouldn't expect to seamlessly have everything move from the internal storage to the external storage, but caring older files and larger files on a sometimes disconnected drive is a long held idiom for working with computers.