Sound is nothing more than the movement of air. For example, the sound of a firecracker going off is actually an experience of shock waves that move through the air due to the rapid expansion of the firecracker. The firecracker expands, the molecules of air get pushed towards you, your ears sense the movement of the air molecules and your brain interprets that as sound. When the air molecules move, there are points in the wave when they are all bunched up together (this is high pressure) and there are points when there are relatively few air molecules (this is low pressure).
Perhaps the easiest way to understand how sounds are recorded is to imagine Thomas Edison's phonograph, the first recording device. To record the sound of a voice, the user would speak into a large horn, at the back of which was a diaphragm (a sheet of flexible material). The diaphragm was connected to a needle, which was placed against a rotating tinfoil cylinder. This is how it worked: The user's vocal cords vibrated and pushed air towards the diaphragm. The flexible diaphragm vibrated, causing the needle to move and scratch a groove in the tinfoil cylinder. To play back the sound, the user just reversed the process: The tinfoil cylinder was rotated, causing the groove to vibrate the needle. The needle then vibrated the diaphragm, which pushed air in the exact same manner as the original voice had.
To record analog audio of a sound today, we use a microphone, which has a diaphragm just like the one in Edison's phonograph. Only this time, the diaphragm is not hooked up to a needle. Instead, the diaphragm in a microphone is connected to a device called a transducer. When the diaphragm is vibrated by the sound of a voice, the transducer converts those vibrations into electrical energy. The portions of the sound that are high pressure cause the transducer to create positive voltage, and the portions of the sound that are low pressure cause the transducer to create negative voltage.
So how do we convert those voltages back to sound? We simply reverse the process, just like we did with Edison's phonograph. We can imagine those negative and positive voltages as waves moving down the microphone's cable. Those waves travel through the wire until they reach a speaker, which is also a type of transducer. This time the transducer converts the electrical energy back to acoustical energy (sound). When positive voltages reach the transducer, the diaphragm in the speaker causes a lot of air to move (high pressure). When negative voltages reach the transducer, the diaphragm pushes less air (low pressure). The process the speaker goes through is the reverse of the process the microphone goes through.
Recall that Edison's medium for storing an audio recording was a tinfoil cylinder. The grooves on the cylinder were a representation of the sound that created them. A tape recording is a representation of sound that is in the form of magnetic particles arranged along a strip of tape. A vinyl record is a representation of sound that is in the form of grooves on a disc. A tape player and a record player simply convert those representations into electrical energy, which passes through the speaker wire to the speaker's transducer, which converts the electrical waves back to sound.