I took an extremely interesting intro to CE course, learning about transistors/memory/logic gates up to LC-3 assembly. Something that was really interesting to me was learning about how a processor's frequency is its speed, determined by the clock signal (if anyone can point to a link of how a clock cycle is made/works or explain, highly appreciated) but my main question is about his claim that any turing-complete computer can accomplish the task of another - performing the same binary operations, just in different amounts of time - that the processor doesn't matter. This depends on memory, though, right? A modern video game wouldn't be able to run on a 64 KiB computer, so I'm wondering if when the textbook and my prof make that claim, it's simply referring to the frequency of the clock cycle (and assuming unlimited/uncapped memory). Thanks!
The key considerations are infinite memory and unlimited time. If those two considerations were met, then all modern computers are equivalent. Since no real computers have infinite memory, and no real humans can afford to wait forever for a calculation, that equivalence is more conceptual than real.
The concept of Turing Complete is purely about data manipulation rules (i.e. a computer's instruction set), not its memory capacity. A machine is said to be Turing complete or computationally universal if it can be used to simulate any Turing machine. A Turing machine is conceptual and has infinite memory. All real machines are necessarily constrained by actual memory.
Your question is answered directly in the Wikipedia entry on Turing Completeness in the section Comparison with real machines
The proposition put forward in the class/lecture you have attended is that both the LC-3 and a modern processor both have instruction sets capable of implementing a Turing Complete machine - what both lack however is the infinite memory and infinite time. Instruction sets and higher-level programming languages are often described as Turing Complete in this context.