There are generally two different types of "standby" travel.
The first is when you have a confirmed reservation on a later flight, and you "standby" on an earlier flight in order to get to your destination earlier. Many airlines allow this (sometimes free, sometimes with a fee) - but I suspect that's not what you're referring to.
The second type of standby is where you have a specific "standby" ticket - which means that you do not have a confirmed reservation on any flight. If there are seats available on the flight shortly before departure then standby passengers will be given those seats. I'm guessing this is the type of standby you're referring to!
Historically a number of airlines allowed for this latter type of standby tickets, often at a discount to the normal price, but with the catch that you might not get on a flight for days or even weeks, depending on how full the flights were.
Over the years most airlines have dropped standby tickets - at least to the public - and I'm not aware of a single airline that offers them on US<->Europe flights. Most airlines do still allow standby tickets to their own staff, and occasionally to other groups such as family members of staff, staff from other airlines, travel agents, etc - but not to the general public.