In the context of transfer functions, what is the relationship between the terms "proper", "causal", and "realizable"?
Causality is a necessary condition for realizability. Stability (or, at least, marginal stability) is also important for a system to be useful in practice.

For linear time-invariant (LTI) systems, which are fully characterized by their transfer function, we get realizability constraints on the transfer function. For continuous-time LTI systems, if we work at frequencies for which the lumped element model is valid, we require the system's transfer function to be rational for the system to be realizable. Also for discrete-time LTI systems we require rationality of the transfer function, which implies that the system can be realized by adders, multipliers, and delay elements.

For an LTI system to be causal and stable, its poles must lie in the left half-plane (for continuous-time systems), or inside the unit circle (discrete-time systems). From this it follows that the rational transfer function of an LTI system must be proper, otherwise you would get one or more poles at infinity, causing the system to be unstable (or non-causal).