Most controlled airports have a radio broadcast of current conditions including IFR arrival and departure runways, ceiling, visibility, surface winds, barometer setting, and any unusual conditions. ATC expects arriving and departing pilots to listen to that recording prior to contacting either ground control for departing pilots or the tower/approach controller for arriving flights. It saves radio time and avoids the controller having to transmit all that info to each flight.
Because these conditions can change fairly often, the recordings must be updated fairly often. Each such update is given a letter designator to differentiate it from the others. When a pilot transmit the "with Juliet" phrase, he is telling the controlller that he/she has all the information contained in the broadcast labeled "Juliet" and will comply with any instruction it contains.
When you look on a VFR sectional, the radio frequencies chart will have a column labeled ATIS. If the airport in question has such a broadcast, the radio frequency for it will be listed. It is also shown on IFR charts.