Can I play against or with Xbox 360 players through my Xbox One?

Is it possible to play against and with Xbox 360 players through my Xbox One?

This depends on two main factors:

I am running an Xbox One version of the game on my Xbox One

No, in almost every game the multiplayer functionality does not support cross-platform features.

There are very few exceptions, such as World of Tanks allowing for cross-platform gaming.

I am running an Xbox 360 version of the game on my Xbox One

Yes. This is through the backward compatibility feature. More details can be found from the official site here:

Xbox One Backward Compatibility

Here is a relevant question from the feature's FAQ:

Can I do multiplayer gaming with friends who are playing the game on Xbox 360 or Xbox One?

Yes. For games that offer multiplayer gaming, you'll be able to play with others who own the title on Xbox 360.

The list of supported games can be found on the official site here:

Supported Games

Yes, my friend has xbox one and I have xbox 360. I just played Samurai Shodown II and NBA JAMs On Fire Edition with him on Live.