The specifics vary a bit around the world, but the basic call anywhere is a position call; this is a radio call on a common frequency or "in the blind" to alert listening aircraft to one's position. It will include the party addressed, identification, location, generally from a known reference point, one's altitude, and direction of flight or intentions. If approaching a uncontrolled field to land, for example, one might say "Coolaroy traffic (party addressed), Cessna 12345 (identification) is ten miles south (location) at five thousand feet (altitude), northbound (direction), landing Boogan (intentions)."
Operations to and from runways will use standard terminology about movement on the surface or in the air, using the same basic format as a position report. This may include position in the traffic pattern (downwind, base leg, final approach, etc, or taxiing or taking off.
Calls may include calls to other traffic to to coordinate. Traffic may announce a longer downwind leg to accommodate another airplane, or coordinate the entry into a traffic pattern. Aircraft may also call out identificatin of another aircraft, to put everyone on the "same page," or in other words, to enhance understanding and clarity.
When in doubt, it's best to be as clear as possible. I can't tell what the registration number is on another aircraft at any distance I'm likely to see in flight, but I may be able to tell the color. If the other airplane says "Bisquick Traffic, red Cessna 469Z is left downwind for runway 18, it helps me identify which aircraft on the downwind is talking; there may be several. If see the red one, I have a better idea. Conversely, I might ask, "are you the red one?" A little plain communication goes a long way. The general rule is to minimize conversation and excess language: don't use any more radio time than necessary, because it may be the words you miss that are the most important ones spoken. Someone might have called a position or intention or even an emergency during the time someone was talking too much. The general conventions and concise use of the radio are fairly universal.