Independence of latent price and market microstructure noise When examining how Market Microstructure works and affects price formation, there is talked about the: 

"Assumed independence of of latent price and microstructure noise" 


From "On the Correlation Structure of Microstrucure Noise: A Financial Economic Approach", Diebold and Strasser, 2013. 


I am struggling to understand what this means exactly? 

For example, when examining a lot of financial trades and quotes, and looking for a underlying latent price - what does it mean to assume independence of the noise? And what do you assume the noise is then specifically? "Wrong" trades? Computer-errors?
I guess you're asking three inter-relatated though conceptually separate questions:


What is the latent or efficient price?
What is market microstructure noise (as opposed to non-market-microstructure noise)?
What does it mean to assume that the efficient price process and the microstructure noise process are independent?


I won't attempt to provide a full examination of the issues raised by these questions: entire articles could (and have been) written about them. I'll only attempt a summary review.


The efficient price process is, by assumption, unobserved. Nevertheless, under the maintained assumptions of frictionless markets, perfect and symmetric information, and rationality of all market participants, one can apply some basic theorems to infer some statistical properties of the efficient price process. Most importantly, the efficient price process must generally be a semimartingale with respect to some filtration. In a continuous-time setting, it is also straightforward to show that the continuous (i.e., non-jump) component of the efficient price process must be local Brownian motion with local drift and variance components.

If the observed price process isn't a semimartingale, it's either because there's an additional component (usually, but not always, called market microstructure noise) or because the maintained assumptions (rationality, perfect and symmetric information, etc) aren't valid to begin with. Let's pursue the first reason.
To make market microstructure noise more than just a catch-all (and tautological) component, it's important to relate any observed or known deviations from the semimartingale property to actual properties of the ways in which trading is conducted. I'll mention three such phenomena (there are several more, of course):


Trading is known to produce not just one price, but two prices: the bid and the ask price. The bid-ask spread is in general non-zero even in a competitive market, for reasons well-explored in the literature of the past 25 years. (In econometric practice, one frequently calculates the average of bid and ask prices and tries to model this average as some kind of martingale but, strictly speaking, that's not always valid.) 
Because of inventory and liquidity effects, there's often predictability of price movements at the very highest sampling frequencies. E.g., if prices are observed every second, it may be possible to predict price movements out to a few seconds. This predictability is at the heart of what algorithmic trading rules are all about.
There may also be some mean-reversion of prices and volatilities at the very highest sampling frequencies due to the so-called "bid ask bounce": Empirically, it's been found that if a given trade happens at the bid, the very next trade is more likely to happen at the ask than at the bid, and vice versa. Because there's a spread between bid and ask prices, there's a bit of extra volatility that arises at the very highest sampling frequencies. This usually shows up in so-called volatility signature plots, which plot the intraday sampling freqency $\nu$ against the associated realized volatility $\sigma_\nu$: At the very highest sampling frequencies, $\sigma_\nu$ is frequently an increasing function of $\nu$.


Note that these "noise components" are generated as a consequence of the fact that trading takes place in real places among real people. As such, these noise components are not akin to those generated by computer programming errors. I.e., flash crashes in certain stocks are not an example of market microstructure noise.
To assume, then, that the latent or efficient price process and the market microstructure noise process are mutually independent really means that they're generated by two underlying processes which don't interact with each other. E.g., if new information enters the market, it should affect the value of the semimartingale but not meaningfully alter the noise process. (The interaction between the two processes is usually assumed to be additive, but I suppose one could generate a statistical theory that works with multiplicative rather than with additive noise.)

Naturally, it is not possible to verify directly that the independence assumption is correct, as one of the two underlying processes isn't observed. However, if it were the case that some observed departures from the semimartingale property are such that they cannot be explained usefully/credibly as being related to various market imperfections (such as the bid-ask bounce), one would have reason to doubt the empirical usefulness of the entire approach to modelling observed prices.