It is possible to digitise real world sounds with an ADC (analogue to digital converter). A sample is then taken of the amplitude of the analogue wave form. (Tascam and Motu use adc and dac, if you are a hip hop or jazz producer this article teaches and helps you learn the basics)

Frequencies are made by play back of the ‘sequential sample amplitudes’ at a specified rate. The recording is determined by the sample rate which determines the frequency response.
The sampling rate can only reproduce frequencies that are half of the sample rate (Nyquist). Foldover could occur when sampling records frequency above the Nyquist frequency. These frequencies may be shown at lower frequencies when recorded.
ADC will normally use filters to prevent such frequencies from being recorded and when these waveforms are being produced synthesizers users will normally want band limiters of samples so that fold back is not an issue.
The sample is then quantized to give it a value that represents the amount of bits available for each sample (The number zeros and ones). When a sample is quantized, the amplitude of the sound wave is rounded to the nearest available digital value (approximation). Less bits more approximation. There is an approximation error which is the difference between the analogue value and digital value.
Large approximation error equals large quantizing noise being produced. Larger sample sizes or bit depth reduces the amount of noise, corresponding to the dynamic range of the system.

Hip hop producers will not need to understand this level of detail but sound designers and vst plugin writers will.

Modern Samplers and Synthesizers have over-lapping functionality, which leads to confusion by beginner jazz and hip hop producers. Samplers don’t have synthesis generation capabilities of oscillation for wave creation or other modelling types. A pre-recorded digital audio file sample is used as sound source synthesis type filtering, gating or modulation is utilized for sophisticated effects processing with digital audio file editing that changes the sound of the sample.

A Sampler is either stand-alone hardware, which is normally part of a musical instrument/dj equipment or software. The sampler gives the opportunity to record a sound formed in a number of ways in a digital format. Using principles similar to those mentioned at the start of this document. The sample file can be a musical recording, natural occurrence, or even electronic sounds produced by a synthesizer. Samplers increase the quantity of instrument sounds and effects that music makers can work with. Samplers are purchased with pre-recorded samples, and some samplers can also be interfaced with MIDI keyboard/controller with the digital out routed to a hardware/software mix or analogue out to audible speakers. The sample can then be utilized in a sequencing project (MIDI and digital audio tracks combined) or imported into a waveform editor for further editing. Modern sampler models are often capable of up to 128-note polyphony. E.g. if a Piano sample is loaded on the sampler then several octaves of piano samples are loaded. Trigger of notes can be then be achieved with a MIDI keyboard if MIDI compatible). Some samplers come with an integrated sequencer that allows the creation of patterns with the samples. The Sampler outputs in digital data stream can is normally routed into a Mixer or directly to a computer with a multi-track audio application and can be recorded as a track in a project.

Synthesizer is used to reproduce sophisticated sounds for midi instructions or digital audio. These sounds are often based on recorded instrument sounds or synthesized sound. Important features of synthesizers are Multi-Tumbrel and Polyphony. They enable sequencers to create many tracks (of different base sounds) with each track having a sequence of notes playing simultaneously.
Multi-timbral synthesizer has the ability to produce more than a single instrument sound at a time.
Polyphonic synthesizer has the ability of playing more than a single note at a time.
Polyphonic and Multi-timbral are the basic concepts of the output of the synthesizer the way this is normally achieved is mentioned below by FM or wavetable synthesis.
Synthesizer originally used electronic circuits to generate tones. The performances of these circuits are mimicked by digital circuits in hardware or by digital modelling in software applications.

• VCO (Voltage Controlled Oscillator)
• VCF(Voltage Controlled Filter)
• VCA(Voltage Controlled Amplifier)

Envelop Generator

Envelop Generator control and shape the signals of these devices. It isolates certain aspects of the amplitude envelope to allow modification. Attack, Decay, Sustain and Release. The time it takes for these to happen can be modified affecting the eventual sound.

Soundcards use the data received by Sequencer to produce sounds.

Frequency modulation.

Past methods of synthesis include frequency modulation. Sound cards and computers would reproduce instruments sounds through algorithms, mathematical representation and electronic means. Electrical means were done by oscillators producing sine waves. FM synthesis required many oscillators. The waves are mixed to represent complex sounds (Modulator waves mixed with a carrier wave). This is achieved by the amplitude of the modulator waveform to adjust the frequency of the carrier wave. This is done with an envelope generator using the frequency and amplitude of the modulator wave to change the timbre of the sounds produced.

Wavetable synthesis

Wavetable synthesiser produce sounds based on stored digital samples of musical instrument sounds(Jazz and hip hop samplers). They are played back by digital oscillators the pitch is then manipulated by filters and envelope generators. The samples, hopefully recorded at 44.1 kHz / 16-bit Resolution, combined with effects and processed by the DSP capabilities of the soundcard. The frequency of playback is set to obtain the right pitch. Not every note of an instrument need be recorded to reproduce the right pitch. This achieves a reduced amount of memory being used. Wavetable synthesis has a better sound quality frequency modulation. There are several types and approaches to sound synthesis: Synthesis, Additive, Physical Modelling, Harmonic Synthesis, Sampler Synthesis and Granular Synthesis. These are the main forms
Synthesizers produce sound by generating a wave pattern and then allowing for the manipulation of the wave form by the various modules of the synthesizer.