This is a big topic and the most suitable answer depends on your background. Since you're a mathematician I assume that you are familiar with representation theory. I'll try to indicate the key rep-th terms corresponding to the terms in the OP, and hope that helps.
For the basic example one starts from the Yangian of $\mathfrak{gl}_2$. It has a presentation (called Drinfeld 3rd realisation = Faddeev--Reshetikhin--Takhtadzhyan presentation = '$RLL$ presentation') by generators and relations. The generators are often combined into an operator $L(u)$ on an ('auxiliary') vector space (which is 2d for the case of $\mathfrak{gl}_2$) with entries that are formal power series in the 'spectral parameter' $u$ whose coefficients lie in the Yangian. This is the $L$-operator, sometimes called monodromy matrix -- though that name often also/instead denotes its image in a representation, see below -- and also often denoted by $T$ instead of $L$. The generators are subject to quadratic relations called the '$RLL$ relations' because of their symbolic form (and called the 'fundamental commutation relations' by Faddeev), where the (rational) $R$-matrix contains the 'structure constants' of the Yangian. For this to define an associative algebra, the $R$-matrix needs to obey the Yang--Baxter equation.
Any finite dimensional representation of $\mathfrak{sl}_2$ gives rise to a representation of this Yangian called an evaluation representation, with an 'inhomogeneity' parameter that can be either viewed as a complex parameter or an indeterminant. In the basic case we do this for the 2d (spin 1/2) irrep and take trivial value of the inhomogeneity parameter. Physically, this is a single site. The image of the $L$-operator here is sometimes called the (local) Lax operator. Bigger representations can be constructed by taking tensor products to obtain the Hilbert space of the spin chain (multiple spins). The image of the $L$-operator in the resulting space is the (global Lax operator or) monodromy matrix. It still acts on the auxiliary space as well. Taking the trace over this auxiliary space gives the transfer matrix, which is the image of an abelian subalgebra of the Yangian (called the Bethe subalgebra) which, when expanded in the spectral parameter, provides a family of commuting operators (conserved charges) on the spin-chain Hilbert space including the translation operator and the Heisenberg XXX Hamiltonian. Finally, in this language, the algebraic Bethe ansatz is a sort highest-weight construction of the eigenvectors of the transfer matrix (and thus the spin chain) that produces actual eigenvectors provided the spectral parameters involved solve the Bethe-ansatz equations.
NB. For the XXZ chain, start from a quantum affine algebra instead, with trigonometric $R$-matrix, and everything carries over (in the generic case). For the XYZ chain, start from an elliptic quantum group, with elliptic $R$-matrix; then you can still construct a transfer matrix and obtain commuting Hamiltonians, but the construction of eigenvectors is much harder since (depending on the type of elliptic quantum group) there are no highest-weight representations (spin-$z$ is not conserved).
