By definition, a socialist economy is one in which we don't produce enough so that even within reason, people can have all the stuff they want. A socialist economy, is, hopefully, moving toward such an abundance economy (which capitalism can't do after a certain point), but our consumption is still restricted. Because consumption is restricted, we have to account for what each person consumes in considerable detail. This means we have to attach consistently determinable numbers to a lot of things that people consume, and use these numbers to determine exchange from producers to consumers. Thus, goods and services have prices. "Prices" in this sense follow only from the fact that we do not yet have enough stuff and must therefore account for our consumption. Furthermore, it makes absolutely no sense to just hand everyone exactly the same basket of goods and service, so whatever method of exchange we choose, we have to take into account people's preferences, which we can know only when people make actual choices.
If we call the above a market, then we have defined the term so broadly as to be useless, at least for any philosophical or rational use: in this sense "market socialism" is just socialism.
However, this broad definition does have an important rhetorical use: Socialists are against markets, but markets encompass every reasonable method of distribution and exchange that could work for more than a handful of people. Thus, socialism is unreasonable; therefore capitalism. In addition to the fallacy of the excluded middle (between unreasonable socialism and capitalism), this argument rests on an equivocation fallacy. The market under capitalism does not have the very broad meaning in the first paragraph.
Capitalist markets mean a specific kind of number, determined by a specific procedure, is a price, and capitalist prices have a specific way to motivate choices. The equivocation between the broad and specific definition lets capitalist apologists argue what Dagood aptly labels hopping from foot to foot. On one foot, socialists argue against specifically capitalist markets, so they are against markets. Hopping on the other foot, the broad definition of markets encompass everything, so socialists are against any form of number- and choice-based exchange and distribution, which is absurd.
But really, a socialist form of exchange is not rocket science, and there are any number of different solutions. One super-simple solution is that we estimate the amount of socially necessary labor time (average or marginal) for each good and service we produce, a tractable process. These numbers are the "prices" of the various goods and services. Second, we have some sort of democratic social process such that each person gets a number — which might or might not vary from person to person — every period (month, week, two weeks, whatever); that number is the amount of socially necessary labor time they consume in that period. They go to the store, they buy hamburgers, buns, and toilet paper, and the socially necessary labor time for the toilet paper is deducted from their periodic allocation.
Coordinating the actual production process is more complicated, connecting exchange and production is at least conceptually simple: if we consistently have a lot of good X unsold, we make less of it; if we consistent run out of good Y, we make more.
The above is a "market" in the broad sense, but it is not a capitalist market. First, labor and labor power are not commodities. Second, the socialist prices above are relatively concrete, as opposed to more abstract capitalist prices. Firms do not arbitrarily set prices to maximize profits; prices reflect actual costs of production.
All the above really does is let people do math to do what we economists call budget-constrained utility maximization, which is an ineluctable consequence of using math to make people as happy as possible under external limitations, regardless of the institutional forms of exchange and production.
Socialist exchange is trivial. We don't have to solve massive linear equations. Capitalists (and, sadly, any number of soi-disant socialists) make a lot of noise about the "calculation problem", but it's just not an issue, at least not in the sphere of exchange. One key is that we never have to calculate the whole economy from first principles. An economy is dynamic, and we already have a set of "prices" and "social allocations of consumption"; all we ever need to do is adjust the status quo, and if it's too difficult to do it quickly, we do it slowly.