MP3 is a digital sound format, and digital waves are represented as square waves,
like each step in a staircase.

Sampling Rate

The higher the sampling rate, the more steps used to approximate the curve,
and the closer the digital signal sounds to the original sine wave.

ie:

When you are using a digital format, it is always an approximation of the analog sine wave, so a higher sampling rate will improve the sound quality, up to a point.
There is a point where our ears cannot detect the difference anymore, because the sounds are outside of our hearing range.

Commonly seen sampling rates are 44.1KHz, 48KHz, 96KHz etc

Frequency is measured in Hertz, or cycles per second. 44.1KHz,
is 44100 cycles per second.

Bit Depth (commonly 8 bit, 16 bit, 24 bit or 32 bit)

Bit depth is different, but also related to digital sound. Using a higher bit depth results in more precise sound calculations.

When you are merging, mixing, multiplying or synthesizing sounds, there are a lot of floating point calculations going on, ie lots of decimal places.

Each individual sound is an approximation of an analog sine wave, so if the calculations are not precise enough, distortion, or unwanted noise will appear in the signal.

The higher the bit depth, the more decimal places are used in the calculations, and the more precise the calculations will become, meaning less noise in the signal. Signal to Noise ratio (SNR) is related to the bit depth. it is measured in decibels dB.

Bit Rate (more info here)

Bit rate is measured in bits per second, most commonly kbps, for MP3 files.

The higher the bit rate, the better the sound quality, up to the point that our ears can detect it.

If you compare 320kbps or 192kbps MP3 files, they often sound similar to most people, but it you compare 320kbps to 128kbps, the 128kbps MP3 sounds tinny, hollow, lots of treble, little bass, like a radio with small speakers, versus a surround sound system.

A higher bit rate results in a larger file size, and better sound quality.

Lower bit rates are achieved by subtracting some of the frequencies in the sound, starting with frequencies outside our hearing range. This is why a reduction from a high bit rate to a slightly lower one doesn’t make a huge difference for most people.

It is only once you start culling frequencies that provide depth or audio richness in the sound, that you start to notice the loss in overall audio quality.

If you reduce the quality too far, the music sounds like you are listening to it through a telephone, which is significantly worse than the CD quality or live reproduction.

Digital representations sound or feel different to true analog sine waves, so a process called oversampling is often used to reduce distortion.
You can read about sampling, bit depth and oversampling in more detail here

Reference