Polymerase: grabs nucleotides floating by and sticks them onto the naked single strand. The polymerase for PCR can survive being heated to near boiling temperatures.
Nucleotides: lots of nucleotides to build new complementary strands.
Other salts and chemicals: needed for the polymerase to work.
The length of the primer: longer primers melt at a higher temperature (just like fats or hydrocarbons).
I think some of the nucleotide bases (G & C?) bind more strongly, so if you have more of those, the annealing temperature also goes up.
PCR can amplify: starting with just a few strands of long DNA, you can make billions of copies of a small region of it. It’s useful for things like forensics and medical diagnostics, when you’re starting with only a small amount of the DNA you want.
Restriction enzymes chop up existing DNA, but can’t make more copies of it. You’d want this when chopping up a plasmid, because the restriction enzymes can break the ring. And restriction enzymes are easier to use than PCR, because you don’t need all the heating cycles, and you don’t need to design and order primers.
The restriction enzymes need a few other chemicals to do their job. CutSmart claims to contain the chemicals needed for hundreds of restriction enzymes, so you don’t have to think about which buffer you’re using for the digests.
The sticky ends of the PCR-amplified inserts need to line up with the exposed ends of the digested plasmid.
You can shock the E. coli to make its membrane more permeable to plasmids — either with chemicals, or electricity, or heat.
Speculating a bit (I’m sure other students will cover Golden Gate assembly, so I want to try something more exotic) …
I wonder if you can use CRISPR for plasmid assembly. Like, use CRISPR to cut the plasmid, leaving sticky ends, then rely on homologous joining to make the insert attach to the cut plasmid.