Oyster charges for multiple routes between two places

Fares on the London Underground are charged according to "zones", with some zones being cheaper than others. So in certain instances, yes indeed it's possible to construct a cheaper fare itinerary by avoiding costlier zones such as Zone 1.

At the start and end of every journey, you're expected to tap the yellow card readers to register your entry and exit points. Separate pink card readers are used as route validator to register whether you used a cheaper route. These are present at:

There are pink card readers on or in between London Overground platforms at:

Blackhorse Road
Canada Water
Gospel Oak
Gunnersbury
Highbury & Islington
Kensington (Olympia)
Richmond
Stratford
West Brompton
Whitechapel
Willesden Junction

There are pink card readers on the platform at Rayners Lane and Wimbledon Tube stations.

You can check whether you can construct a cheaper fare using the Single fare finder.

As I understand Oyster cards, you swipe them on the station you enter, and the station you leave

This is normally the case for train/tube/DLR travel on oyster. Buses and trams have other arrangements. Also oyster cards are based on NFC, not magstripes so it's more of a tap than a swipe.

As in you don't swipe them when changing underground lines (maybe you do on some stations though?).

Normally touching out ends your journey and if you touch back in you will be charged for a separate journey (subject to daily capping).

However there are a number of locations on the network where "out of station interchanges" are allowed allowing you to touch out and touch back in to a nearby station or to a different part of the same station without ending your journey.

So if there is a variety of ways you could have gotten between A and B and one is more expensive, what are you charged?

Ultimately the system looks at what readers/gatelines you used and makes assumptions based on that. The normal assumption is that you took the fastest/most direct route that is consistent with the readers you used.

The zonal fares system means that the exact route you took is not usually a massive issue, the main exception to this is if you want to save money by taking a route that avoids zone 1. In such cases you should look your journey up on the tfl "single fare finder" and use a pink validator if needed.

National rail services that take oyster but are not run by TFL (and the Heathrow branch of TFL rail) can also have higher fares than underground services. This is mostly handled by splitting stations into areas with separate gate-lines, though the system does have some flaws.

(I've heard something about pink validators?)

Yes, some stations have pink validators. You touch your card on these when interchanging to tell the system where you changed trains. The "single fare finder" will tell you when and where you need to use them.

Note: when using the single fare finder be careful what station you select. In particular the rail and underground parts of a station are often considered to be seperate stations by the fare finder.