My sky-high level understanding of modern cryptography is as follows: there is an algorithm and a key.
Sure, a key is generally generated for a specific cryptographic algorithm that takes a key.
The key or key pair is a randomly generated number (I think).
Not necessarily. AES and Serpent just take a key consisting of random bits. RSA uses two random primes that are used to create the key pair, where the keys consist of multiple parts, at least an exponent and modulus generated from these primes. Or it could be a random in a specific range for the private key of ECDSA or ECDH, where the public key is actually a point on the curve. That private key is probably closest to "a number".
The algorithm describes transformations to be applied to the plain text in order to encrypt it.
If that algorithm is a cipher, then yes.
The key is a specific piece of information that is needed by the algorithm to complete this process of encrypting the plain text.
Not sure about the emphasis in that description of the key, as there are other possible pieces like an IV that are required. It's a bit more than that; it is the piece of information that is at least required to complete the process.
What I do not understand is how, precisely, do the algorithm and key interrelate.
That depends on the algorithm, the algorithm specifies what kind of keys are acceptable, and how large they may be (if there even is a maximum).
One can look at a schematic for, say, Serpent and see lots of arrows, boxes, feedback loops and so on. To an interested lay person such as myself, I cannot see what the key (i) does to that set of processes or even where it fits in; (ii) how it is employed such that its uniqueness effectively 'locks' the cipher text.
That depends entirely on the algorithm. In block ciphers such as serpent there is a lot of transposition and diffusion going on, generally trying to create an avalanche effect so that many bits is affected even within a single round.
However, with the asymmetric RSA cipher the private key is used for modular exponentiation during decryption and in ECDSA for point multiplication during signature generation. Exponentiation and point multiplication are more mathematical constructs than the bit operations that are common in symmetric ciphers.
Finally, I understand there isn't a single response to this inquiry given there are different types of keys used for different purposes. In the interests of addressing this, please assume I am referring to the encryption of plain text using a symmetric key algorithm that was fit-for-purpose in 2001 or later.
I've tried to explain it in the simplest way possible, also showing that it matters quite a bit what kind of algorithm we are talking about.
I'd study a bit more about block ciphers and try and e.g. implement S-AES (or SimpleAES, I'm not sure if there is an official name) if you're more of a developer type that learns by acting.