A basic answer is, a sequencer is any program that can record and/or playback either MIDI performance and/or analog/digital recordings.
Traditionally, a sequncer would record MIDI notes (a performance - not a sound) and then send that performance to one or more MIDI sound modules, to trigger sounds.
A smapler, is a device (hardware or software) which can sample sounds (or a preformance of sounds) and play back those sound - either as a preformance "loop" or simply sounds triggered by some type of controller (ie: a MIDI keyboard).
A synth, creates sound using electronic wave forms, filters, etc. to create sounds.
If you have a harmonica sample in a sampler, and want to use the keyboard of a synth to "trigger" that sample you would take a MIDI cable out of the synth (the controller) into the sampler (the slave).
Keep in mind, if you have a sample in a given format (say Kurzweil) you may not be able to import that sample into say a Roland sampler. You devices support various formats and some do not.
Some synths have various sample record or playback abilities, and some don't. Some samplers allow you to change sampled sound significantly, and some don't. Some may disagree, but I've always felt, if you are looking for realistic accoustic instrument sounds (piano, sax, etc) you would likely do better with samples and if you like to create original sounds, unique to your creative impulse, a synth would likely be better.