Synthesizers are, internally, an interconnected collection of sound creation and processing "modules".  Synthesizers can imitate "real-world" sounds like "flute" and "kettle drum" or create new, "impossible" sounds that are only heard in audio CDs and movie soundtracks.  They create sound by modifying an oscillator's (also called "tone generator") simple waves with filters, envelopes, amplifiers, wave layering, etc.  Synthesizers can be connected to PCs and other synthesizers using MIDI (Musical Instrument Digital Interface).  The sounds are triggered by someone playing the synthesizer's keyboard, by it's sequencer, or by an external triggering via MIDI.

Samplers can record any natural sound with an attached microphone, which may then be converted and used as a waveform/sample.  The wavesample is assigned to a given key on the keyboard, and the adjoining key sounds are made by "pitch-shifting" (speeding up or slowing down) the sample.  The more the sample is "pitch-shifted", the less "natural" it sounds, so several samples of the sound may be recorded at lower and higher pitches in order to reduce the amount of pitch-shifting needed to map all of the keys.  

The difference between synthesizer and sampler becomes blurred, as samplers usually include synthesizer circuitry to further modify their samples with filters, envelopes, LFOs, etc. 

A keyboard sequencer can be a simple "Record" button that saves a performance in real time.  More advanced "workstation" keyboards often include an 8-track MIDI recorder, where different instrument sounds on different tracks may be "overdubbed", creating a full band arrangement.  Sequencers can play back music, sending different tracks on different MIDI "channels" to MIDI-connected keyboards, modules, and computers.  The "workstation" keyboards with sequencers usually include a synthesizer and wavesample banks, and occasionally sampling capability as well.

Some synthesizer, sampler, and sequencer modules don't include a keyboard, and are triggered via MIDI.

A MIDI keyboard connected to the computer is normally used to trigger the computer's soft synths, samplers, and sequencer.

Computer soft synth and sampler programs are often as feature-rich (minus keyboard), but are significantly cheaper.  

MIDI sequencer programs on the computer far outshine the keyboard workstation versions, with graphic timelines, note and parameter editing capabilities.