Hello, I have a laptop that doesn't have much storage, but i have a 2T external HDD.
If i download a program/application on the external drive, do I have do download any type of file locally on my actual laptop, or can everything remain on my external drive and whenever i want to use that program/application i have to have the external drive plugged in to my laptop?
*Assumption: Windows*
You probably meant installing, not downloading.
Yes, this generally works, because the installation process doesn't differentiate drives as "internal" or "external".
However, there are some consequences:
1) you always have to mount the drive under the same letter
2) ideally, you shouldn't edit those files when mounting the drive on another machine
3) software can be referenced by another one (you won't even know it's being used) - this can lead to errors and unstability (even BSODs)
4) when the drive isn't connected, installation will be treated as corrupt, so you have to be extra careful with software that cleans the registry or removes leftover shortcuts (e.g. CCleaner).
It's always a better idea to keep all programs on the "always on" drive and move data to the external one.
ISSUE!
A lot of software will use your user account folder (C:\users\<you>) or the C:\ProgramData to keep data by default - no matter what installation folder you choose during setup. Sometimes there's no way to configure this.
This can lead to a situation when not only your space gain will be smaller than you expect, but more importantly: even when you unmount the external drive and lose the apps, you're still carrying (mostly unusable) data.
Also, the data usually grows over time and the actual software doesn't.
So, apart from the suggestion to use the external drive for data - not executables, there's another one: you should seriously rethinking what you're installing.
Most laptops have at least 256GB and Windows+mandatory stuff (e.g. drivers) are very unlikely to take more than half of that.
Remaining 100GB+ should be plenty of space for software (other than games, obviously) - even if you're hoarding large suites like I do (Visual Studio, Matlab, Adobe Lightroom+Photoshop, MS Office and so on).