Hi
Digital circuits are mainly main up of gates eg. AND, OR, NOT, NOR, XOR, NAND. One need to know the truth tables. eg. for a two input (A and B) AND gate with one output X ... the truth table is as follows
A B X
0 0 0
0 1 0
1 0 0
1 1 1
When using a scope/logic probe and 1 = high (5V for TTL) and a 0 = low (ground or 0V)
Combination of gates allow one to build a Car Alarm etc.
There are also counters/flip flops/shift registers ... counters (together with a 555 timer or RC connected to the counter) were used for delays, flip flops where used for switching and the PIC RRF and RLF do the job of a shift registers.
Logic Analyzers were using for fast square waves.
Hope this helps
Mr Sparkle