There are several different technical solutions to microphones which influence what they measure. I will limit this discussion to two types: an omni pressure sensitive mic and a figure 8 velocity sensitive mic.
First the figure 8 microphones which does not measure air pressure but instead air velocity: Imagine a very thin not very wide ribbon or band set in more or less free air, suspended from its both ends. (This is actually a rather good description of how a ribbon microphone is created. Typically a very thin alumnium band, say half a centimeter wide and 3 centimeters long). The band will waver to and fro depending on the velocity of air "wind". We can call this that the band acts according to local air velocity. As sound is "wind" going to and fro here, the mic then translates this wavering motion into an electrical signal. You can understand that the air pressure, barometric pressure, is more or less same on both sides of the band. The mic will not react to high or low air pressure. This mic will be sensitive to air movement perpendicalur to the surface of the ribbon, but quite unsensitive to air movement over the thin band creating a sensitivity graph looking like the number 8, hence the name.
Now for the second type of mic I talk about here: an omnidirectional pressure sensitive mic. Imagine a thin membrane tensed across a closed cavity. (Again, this is actually the design of mics and some barometers). Increasing air pressure will push the membrane in towards the cavity, decreasing air pressure will allow the membrane to move outwards. Sound is seen here as a local to and fro in the pressure of air. The mic (or barometer for that case) translates this into an electrical signal. For barometers, the cavity may be fixed reference, say a vacuum. For actual microphones though there is a small hole from outside air to the cavity. This will allow the cavity to be filled with air around with on average the same pressure as outside air. In effect the hole will work as a high-pass filter: only air pressure variations with a high enough frequency can be registered by the mic. The "barometric" pressure, of a very low frequency, will hence be filtered out and only the higher frequency sounds will remain. (Now, some measurement mics may go below 1Hz as lower frequency, but mics used for recording music usually starts at something like 10Hz as there is very little music in lower frequencys). The pressure measuring mic is more or less omni-directional on lower frequencys (same sensitivity in all directions) but will become more and more directional in higher frequencys (say above 1kHz) due to the exact design as the membrane does point in one direction.
As a side note: most microphones used for voice or music today have a cardoid pattern. This can be though of, and sometimes created, a combination of omni + a figure 8. Exactly how to do this is left as an exercise for the reader ;-) .