The WPS button on my modem

Hi, what is the brand and model number of your modem?

It's WiFi Protected Setup.  It's a convenient way to connect your wireless devices to your router.  Of course, any convenience means less security.

In your case, with the button, you press the WPS button and then can connect a wireless device without entering your network key (encryption key, security key, network passphrase, network password, etc).  It usually times out after a few minutes.

There are other variations of WPS.  Instead of entering the network key on a wireless device, you could enter a PIN.  Another variation is instead of a physical button, the router may provide a software button on the connection screen.

Although entering a strong, effective network key, such as 1L1k3J3llyB34ns4ndC4k3!, is an inconvenience, you should require all devices to enter it...to include unauthorized devices.  If you can, you should disable WPS on your router.