Will a 90 watt power adapter (instead of the 60 watt adapter it came with) harm my Inspiron B120? It's made by Dell, and it fits…
No, the wattage listed on the power adapter is the maximum that it can deliver not that what it will. The laptop will only draw the power that it needs. Think of it as a capacity measure of the adapter.
Now the output voltages of the power adapters do matter - as long as they are the same (or very close) you should be fine.
I assume that both power adapters that you are refereeing to are Dell power adapters so you'll be fine in that case.
It's not the power output that's important from the point of view of damage, it's the voltage.
If the voltage supplied to the computer is correct, then no, the adapter will not harm the computer.
It may be helpful to think of the power supply as providing available power, at a given voltage, up to a maximum.
The computer will take what it needs - if the PSU can't provide enough, then the computer won't work, but if the computer needs less than the PSU could provide, then it won't take more than it needs.
However, you need to check that the output voltage of the PSU is not more than the computer can cope with. To a first approximation, it should be the same as the original PSU, in which case, there will be no problem.