A synthesizer is an instrument or a computer program which produces sounds. There are two basic types of synthesis: additive and subtractive. Old analog synths work by additive synthesis: they add frequencies to a basic wave form (normally sine waves, but also triangles) . Most digital synths and computer synths work by subtractive synthesis: they filter and thus subtract frequencies from a wave form.
A sequencer is a computer program which produces a sequence of (normally) MIDI data which is then passed to an 'instrument' in order to be 'played'. These were once hardware components with a certain amount of memory; a programmer or musician would load the memory with several sequences (like drum patterns) then trigger the sequencer to 'play' them. Since the 1990s (if not earlier), sequencers are more often software programs and can be compared to word processors. Instead of writing notes on manuscript paper, one enters them into the sequencer. This makes copy/paste operations very easy as well as facilitating insertions and deletions. Most (if not all) sequencers are able to produce output from the entered notes, although the sounds obtained depend on the computer's sound card. Most sequencers have the ability to import MIDI data as played on a keyboard; they normally have two types of display - keyboard and piano roll.
A sampler is hardware which records 'real' sounds and then converts them from analog form (what we hear) into digital form (for computer manipulation) by sampling the sound - 96 thousand times a second. The faster the sample rate, the better is the conversion.
Ableton is a DAW: one can synthesize sounds and play them via a sequencer. Pro Tools is basically a digital tape recorder which imports wave files, manipulates them, mixes them then outputs them.
I imagine that a 'beatstep sequencer' is a simple sequencer of the original type which one programs to produce drum sequences.