What is a contact?

Sometimes I'll excitedly tell a non-ham that I've just made a contact with West Malaysia/Chile/a guy in Belgium through a satellite.
But what exactly is a "contact"?

Sometimes, a contact can be a 2 hour-long natter about whatever takes your fancy.
But sometimes a contact is just a brief exchange of callsigns and signal reports (technically the minimum that can be considered a contact (or QSO)).

Scarce resources

There aren't that many satellites whizzing past in space, but when they're overhead, the footprint (what it can see) is fairly large. This means that there are potentially hundreds of stations trying to make a contact in the 10 minutes it takes for the satellite to travel from one horizon to the other.
This means that it's bad form, and indeed pretty selfish to want to discuss last night's episode of whatever through this scarce shared resource.
Satellite exchanges are therefore generally just a swap of callsign and Maidenhead locator.

Not voice?

The contact doesn't have to take place using voice though. Wikipedia has a list of modes used in Amateur Radio.
So the good news here is that if you don't like talking to people, there are still a ton of ways you can make contacts.
In fact, there are even automated digital modes that do it all for you.

Contests

Often, there are contests in which stations aim to make as many contacts as they can within a set period of time.
Obviously, the shorter the contact, the more you can make.

It's the challenge

Often the challenge is simply being able to exchange information with another station - e.g. the fun is in the chase. It doesn't so much matter what you say - it's the challenge of building, assembling and operating the equipment and antennas.

It's not just about distance

Distance isn't how you measure the difficulty either.
For instance, in some circumstances getting a signal 7.6km on 241 GHz is a lot harder than me getting a signal to Brazil on 20m.
The reasons for this are a:, physics - radio waves at 241GHz are attenuated severely by most things - including the atmosphere, and b: there aren't nice pre-made transceivers and antennas that you can buy off the shelf like you can with other bands - you have to make transverters, and waveguides, and all sorts of things. Also, even just the loss in the cable from the radio to the antenna is so high at high frequencies that you have to mount the antenna really close to the transceivers.
So getting this signal 0.44km down the road will have taken loads of work.
Likewise, very low frequencies also present a challenge. For instance, the wavelength of the 136kHz band is 2 kilometers. How do you make an antenna that large that's also the slightest bit efficient? https://www.youtube.com/results?search_query=136khz+band. Also, the frequency is so low that pretty much the only thing you can send is slow-speed Morse (QRSS).

A successful exchange of information

I think the minimal definition of a contact is the successful exchange of information, which is usually the callsign (usually mandated by the conditions of the licence anyway) and a signal report.