Hello, gallivantinginseattle, and welcome to FlyerTalk.
Given your username, I assume you're traveling from the US to Ireland. Does your sister use a manual wheelchair or a power chair?
For the flight itself, she will be able to use her own wheelchair right to the gate of the plane. The gate agent will put a gate check tag on the wheelchair (this is the same thing they do with strollers). Just take a look at it it make sure it's got the correct destination on it (and do the same with your bags when you check them).
At the door of the plane, a baggage handler will take the wheelchair down to the hold. Remove anything from the wheelchair that could come or fall off, especially the cushion, and take it on the plane with you. It won't count against your carry on allowance. Put a luggage tag with your name and contact information on the chair somewhere.
If your sister can walk short distances, at this point she will just walk on to the plane and take her seat. If not, there is a special narrow wheeled chair, called an aisle chair, that she will transfer onto to be pushed to her seat. The folks who do this may not be familiar with the right way to transfer someone with more significant disability, so she (or you) may need to do some educating to make sure she is transferred safely.
Once on the aircraft, if your sister cannot walk at all, there is a similar skinny little chair called the on board wheelchair that the flight attendants can use to help get her to the lavatory if necessary. Airline lavatories are very small, but the flight attendants can curtain off the lavatory area if more privacy is needed to help your sister in and out.
At your destination, you will wait until all the other passengers have deplaned and your sister's wheelchair has been brought back up to the jetway. Check the wheelchair over right away for damage, just in case.
There are a lot of threads in this forum about the specifics of air travel for wheelchair users, including information about wheelchair users' legal rights. There is a search function in the upper right hand corner that will help in searching for threads in this forum.