Last October, I had the honor of making such contact with the ISS. I posted about this on my website here: https://rz01.org/na1ss/
The ISS has radios on board that allow amateur radio operators to send and receive APRS messages, talk to other hams via their built-in FM repeater or to receive SSTV images. They also have amateur TV stuff on board but I have not explored this yet. Crew members with an amateur radio license can pick up the mic of the radio that acts as a FM repeater to make contacts with other hams on the earth.
A good few years ago I had a crack as using the ISS's crossband repeater when it was on and could hear myself clearly with a Kenwood TH-F7E and home-made dual-band crossed dipole.
Unfortunately no-one else could work me, because they'd boosted the satellite's orbit, the TLEs hadn't been updated, and so everyone with a nice motorised antenna positioner was aiming at the wrong part of the sky.
Me standing in my back garden pointing roughly in the direction of the fast-moving bright spot? Nah that worked perfectly :-D
Nice! I made my first LEO sat contact also in the garden with an arrow yagi and a cheap HT. It was fun but I prefer operating the easy way and building a nice station. Some neighbours watched me making this first contact and they now speak to me differently...
I was out struggling to rig up a new antenna I'd made - nothing was going right and I was getting frustrated - when it occurred to me that if someone were to ask what I was doing, they would've seen a rambling lunatic trying to sculpt a pile of coat hangers while yelling "I'M TRYING TO REACH THE SPACE STATION!!!"
The ISS has radios on board that allow amateur radio operators to send and receive APRS messages, talk to other hams via their built-in FM repeater or to receive SSTV images. They also have amateur TV stuff on board but I have not explored this yet. Crew members with an amateur radio license can pick up the mic of the radio that acts as a FM repeater to make contacts with other hams on the earth.
You can contact the ISS via a handheld setup (FM radio and a yagi antenna in your hand) or with a stationary setup like mine: https://rz01.org/leo-sat-ground-station-v3/