What does it mean "non-Walrasian" or "quasi-Walrasian"?
Well non-Walrasian is an model that is not Walrasian so to explain that we should explain what  a Walrasian model is. Generally speaking in macro context when we talk about a Walrasian model it is a Walrasian general equilibrium model.
A Walrasian general equilibrium has the following characteristics see MWG pp 579:

Definition 17.B.1: Given a private ownership economy specified by
$$ \{(X_i, \succeq_i) \}^I_{j=1}, \{Y_j \}^J_{j=1},\{(\omega_i, \theta_{i1}, ... , \theta_{iJ})\}^I_{i=1}$$
an allocation $(x^*,y^*)$ and a price vector $p=(p_1, ..., p_L)$ constitute a Walrasian (or competitive, or market, or price -taking) equilibrium if:
(i) For every $j,y^*_j \in Y_j$ maximizes profits in $Y_j$ that is, $p\cdot y_j\leq py^*_j$ for all $y_j \in Y_j$
(ii) For every $i,x^*_i \in X_i$ is maximal for $\succeq_i$ in the budget set $\{ x_i \in X_i:p \cdot x_i \leq p \cot \omega_i + \sum_j \theta_{ij} p \cdot y_j^*\}$
(iii) $\sum_i x_i^* = \sum_i \omega_i + \sum_j y^*_j$

Slightly different definition holds for pure exchange economy, but for more detail you can have a look at the rest of the chapter in MWG. The gist of what is a Walrasian general equilibrium (and thus if model is Walrasian) is given by the authors in the brackets, it is a model characterized by competitive, or market, or price-taking equilibrium. A non-Walrasian model would be some general model that is not characterized by such equilibrium.
Following Calvo (1978)  quasi-Walrasian model is (emphasis mine):

These [quasi Walrasian models] are models
where agents perceive themselves as infinitesimal micro units, unable to exercise any kind
of &quot;market power&quot;; the agents set the variables under their control in order to maximize
their objective functions (utility, profit, etc.).
In contrast to the so-called &quot;disequilibrium&quot;
or &quot;fix-price&quot; models (see Allan Drazen)
there is no exogenous rationing or price
inflexibilities. However, they differ from standard Walrasian models (for example, see
Kenneth Arrow and Frank Hahn, chs. 1-5) in
that agents are not necessarily price takers
and, typically, markets are incomplete

The explanation above is pretty much self explanatory.
Lastly, New Keynesian general equilibrium models are quasi-Walrasian precisely because they allow for market power (in fact that is what in some NK models creates price rigidities/stickiness), or incomplete markets and so on. You can see nice overview of NK models in Woodford Interest &amp; Prices, you will see that virtually all of them have some twist where there is market power, or similar market imperfection.