What does "Relativistic" mean in Quantum Mechanical Terms?

There is no full consensus about what relativistic means, but as a good start, we can take the criterion given by Tim Maudlin in his paper "Space-Time in the Quantum World":

[A] theory is compatible with Relativity if it can be formulated without ascribing to space-time any more of different intrinsic structure than the [...] relativistic metric.

This directly implies that the laws must be invariant under Lorentz transformations, there is no preferred frame as such and nothing can propagate faster than light. This is a good notion since it does not need things like "speed of a photon" that are, as you pointed out, a bit problematic.

This answer is meant to add to Luke's excellent concise answer, so please read his answer first.

In quantum mechanics, only measurements have the statistical distributions, the "uncertainties" and all the things that are (validly) bothering you. As you point out, this makes notions of measured spacetime co-ordinates problematic.

But the underlying theory that lets one calculate these statistical distributions can be Lorentz-invariant.

It is not emphasized enough, particularly in many lay expositions, that much, if not most of quantum mechanics is utterly deterministic. This deterministic part is concerned with the description and calculation of the evolution of a system's quantum state. Aside from some more modern mathematical techniques and notations, this part of quantum mechanics probably wouldn't look very alien or physically unreasonable to even Laplace himself (whom we can take as a canonical thinker from the philosophy of determinism). This quantum state evolution takes place on an abstract spacetime manifold just like classical physics. When a quantum theory is said to be relativistic or Lorentz invariant, it is usually the deterministic quantum state space evolution that is being talked about. Note, in particular, that no measurement takes place in this part of the description, so there's no problem with a spacetime manifold parameterized by zero uncertainty spacetime co-ordinates.

We model measurements with special Hermitian operators and recipes for handling them called observables. Given a system's quantum state, these operators let us work out the statistical distributions of outcomes of the measurements we can make on a system with that quantum state. When people talk of quantum uncertainty, Heisenberg's principle and all the rest of it, they are speaking about the statistical distributions that come from these measurements. So, whilst Schrödinger's equation (the deterministic, unitarily evolving description) for the electron in a hydrogen atom is written in terms of zero uncertainty space and time co-ordinates (known as parameters to emphasize that they are not measurements), the outcome of a position measurement is uncertain and the position observable lets us calculate the statistical distribution of that outcome.

How exactly did Dirac incorporate SR into his wave equation?

Relativity treats space and time on equal footing. Lorentz transformations ("boosts") "mix" space and time more or less analogously to the way a 3-dimensional rotations "mixes" the usual $(x,y,z)$ coordinates of space.

The Schrödinger equation describes non-relativistic quantum mechanical systems well, but is second-order in spatial derivatives and first-order in the temporal derivatives. In other words, space and time are not on equal footing.

Dirac, being a formal theorist, looked for an equation that would reproduce predictions (e.g., the energy levels of Hydrogen in the limit of slowly moving electrons) of the non-relativistic theory while treating space and time on equal footing. The equation which he discovered and now bears his name is a first-order differential equation in space and time. This "equal treatment of space and time" is the sense in which the Dirac equation incorporates special relativity.