It's most commonly called a cardiac monitor or just a monitor. It takes an EKG of the heart, and can also do 12-leads (and some can do 16-leads). It shows you the patients heart rhythm, and a 12-lead can be useful for recognizing a heart attack
or other cardiac problems such as bundle branch blocks. The sticky things are called electrodes. And yes, they use memory cards to store the information. Some are transmitted straight to a computer, others (like the ones I use) we connect with a cable to our tablets to sync the EKG to. Some common brands you can look up to read more about them are Zoll, Philips, and Lifepak.
And ambulance drivers do not have them. Paramedics have them.