One definition of a split is to transmit on one frequency and receive on another. There are a few reasons to do this. A 2M FM repeater uses split frequency so as to avoid desensitization of the transmitter/receiver circuitry; by this I mean it would attenuate the receiver by a strong signal being transmitted on the same frequency it was simultaneously listening on. On HF, a DX station would use a split frequency so as to avoid interference of those trying to listen for his signal. He operates in split mode by transmitting on one frequency and receiving on another frequency. This way if hams that are in a pileup are still calling him as he wishes to go back to another station; that particular station he is trying to call can hear him. A split is usually announced by him or on a DX packet spotting cluster as "QSX". Most often the DX station will state during his exchanges info like: "UP", "UP 1", "UP 3 to 5", or possibly another frequency like "listening 245". The latter might indicate if he were operating on 40M that he was listening for you or I to transmit on 7245 kHz. In the case of 40M I have worked DX by listening in LSB mode in the CW sub-portion. Other countries can transmit in voice where we here in the US can only transmit in CW. So it is quite common to hear a DX station transmitting for example on 7060 kHz and receiving on 7245 kHz. (The frequencies are picked arbitrarily).
Sometimes the DX station is listening for us on a small slice of frequencies, like when he says "5 to 10 up". That is 5 to 10 kHz up from where he is transmitting. There are different techniques to getting a freq to transmit on. You might first of all want to listen to where you want to transmit. You would do this so as not to interfere with another ongoing communication by other hams. You probably wouldn't like it if you were talking with another ham and all of a sudden other hams start transmitting without any regard for your transmission. You would also be listening for a clear (or relatively) clear frequency. At times this may be downright impossible with many other stations in the pileup. If you're quick enough with changing your VFO's you might hear the DX station working W8RIT and then you would listen at the time W8RIT might be transmitting in the portion 5 to 10 up. If you do hear that station, you would then know that is where the DX is/was listening. Different DX stations have their own techniques; some sweep up and down the 5 to 10 portion listening for the stations they will work. Others may listen for a few minutes on one particular frequency and then move up or down. Some go randomly from one end to another and stop whenever they hear a strong signal.
Everyone makes a mistake now and then. If you make a mistake, you may hear "band cops" say "UP". Sometimes they can be rather unpleasant in their demeanor. Sometimes they create more interference than they're trying to solve. It's only a reflection of that personality, not you. Everyone is human and makes a mistake occasionally. If you feel the need to correct someone in this case, you first might want to think twice...will it really help or will I create more noise? If you do, you may want to state their callsign to attract their attention (not to single them out) that it was them who needs to operate split. You might want to state simply, "W8RIT up please". If you can, transmit at a time when you will not interfere with the DX station as someone else may be trying to get their exchange and you cover it up. It may be a rare thing for them to work a DX station, get through the pileup, only to have you transmit over the top of the DX station. Also, never, ever, transmit where you do not have license frequency or mode privileges. If someone is operating on the wrong split frequency, such as, 7060 in LSB, that does not give you the right to transmit there to inform them. You are then intentionally breaking the law, they are unintentionally doing it.