Changing 7-speed Shimano cassette to 10-speed
EDIT As ojs points out in the comments below, the freehub's width of 7-speed systems mismatches those of 8- to 10-speeds. This means the rear hub/wheel has to be changed as well, in addition to everything listed below. Clearly a cost of such change now would exceed the price of a half new bike.



It would require changing the complete drivetrain and the rear wheel. 

However, the good news are that a freewheel coming with 7-speed Shimano cassettes is compatible with 10-speed systems, so you won't need changing the rear hub/wheel.

The reason why the complete drivetrain has to be changed is that the front chainrings may not be working well with a narrower 10-speed chain. 


The rear cassette has to be changed for obvious reasons to have more cogs
The rear derailleur has to be changed because of a different cable pull, range and tolerances required by a 10-speed system.
The rear shifter has to be changed to have 10 indexed positions and matching cable pull
The shifter cables/housings might be preserved, but they are cheap and it is recommended to change cables periodically to have better shifting experience, so it is highly recommended to get new ones.
The chain is to be changed because 10-speed chains are narrower, and a 7-speed one will be too wide and will jump all over the cassette.
The front chainrings are spaced to match a wider 7-speed chain. There is little possibility it will work nice with a narrower chain.
The front derailleur matches the front chainring spacing.
The front shifter goes away together with the derailleur.


What you need for drivetrain parts is called "a new groupset" — a name that encompasses all the parts I've mentioned. 

Converting the rear wheel to a new hub would require a complete wheel rebuild. Oftentimes it means that buying a new complete inexpensive rear wheel would be cheaper.