Socialism means worker ownership of industry. Full stop. But there are so many ways to achieve this kind of arrangement and even more forms of expression. Socialism doesn't require repression, a one-party state, or even a strong government. Some socialists believe the government should just manage large and powerful labor unions. Some socialists believe there should be any government at all! Anarchists are just socialists with particularly radical aims.
As a few additional notes: socialism and capitalism are
not economic systems or institutions. Capitalism does not mean using marketplaces, allowing the buying and selling of goods, the use of money, or anything along those lines. One of the greatest pieces of intellectual slight that the political right ever accomplished was persuading people this was the case.
Socialism and capitalism are
social systems. Societies are normally divided into different groups of people, with, if not perfectly clear lines, at least relatively clear distinctions between them. These are classes. Capitalism describes a social system where there is a class that has control over the uses of capital (upper class/capitalist/bourgeois), and a class that has no control over the uses of capital (working class/labourers/proletarian). Socialism describes a social system where there is a single undifferentiated class, where control of capital is distributed evenly between peoples. (side note: this is not the only way to divide classes - any division that in society that is relatively persistent and of which the members share common interests. In a racist society, you have black and white classes. In a sexist society, you have male and female classes. Societies can have multiple different axes of class. Marx
did think that all other class divisions ultimately stemmed from economic divisions, but this is not a necessary part of socialist thought and was criticized by many socialists).
This is at least partly orthogonal to the economic system. You can have capitalism that does not have free trade or open markets - for example, companies that create monopolies and practice unfair competition to keep out competitors and concentrate capital in a single company in the market. You can have capitalism that doesn't have markets at all - the state controls all the capital, and the state is dominated by a small class, so that class de facto controls all the capital and are capitalists, and those not in the state have no capital and are therefore labourers - this is a state capitalist society and arguably the Soviet Union was one. You can have socialism that has no markets - think a democratic Soviet Union - although I think in practice this one is very unlikely. And finally, you can have socialism with markets, which is something like a co-operative economy or even just an expanded version of a universal basic income system.
However, certain social systems tend to lead to certain economic systems being instituted, so they're not entirely orthogonal. Capitalist societies, for example, tend not to have particularly free markets after all - it's better for capital owners if they can create monopolies and the like. Socialist societies tend to have certain kinds of restriction on economic activity - if they're market socialist societies, for example, they tend to have very high inheritance taxes to prevent accumulation of capital over time.
Finally, socialism and capitalism are obviously 'ideal' types. They're models, simplified into two categories to help make some of the theories behind them easier to understand. Real life is much more complex (and some theories adapt to cope with this accordingly). We don't live in a 'pure'/perfect capitalist society. Most labourers do actually own some form of capital (for example, people who own their house outright, or have long-term savings), blurring the distinction between the class with capital and the class. We can make comparative statements though: over the last twenty years, there's relatively strong evidence that developed countries have become more capitalist and less socialist. But the key point is that thinking of socialism and capitalism as entirely dichotomous systems is probably the wrong way to think about it. They're a spectrum.
Additionally, even if you think that we ought to be more socialist, that's pretty vague and could lead to any number of different conclusions. Rather than arguing for a sharp transition to some entirely different institutional set-up (revolutionary socialism; or working outside the current system to bring it down), socialism as a political movement and organization can just involve reducing the extent to which our societies are capitalist and increasing the extent to which they are socialist slowly over time - usually called reformist socialism after the German movement. I think this is probably why Sanders classes himself as a socialist.
Social democrat is a confusing term because it used to be synonymous with reformist socialism. Reformist socialism, inside democratic societies, obviously argues for the democratic instantiation of socialist principles, and hence adopted the moniker democratic socialism - or parliamentary socialism in some places (UK, for example, where the Labour Party was a parliamentary socialist party). Democratic socialism, socialist democrat, social democrat - all the same terms for the same thing.
But there was a perception that a lot of social democratic parties stopped being socialist, in the sense that they were no longer concerned about a classless society as the end-goal, and instead wanted something like 'paternalist' capitalism - the idea that it's fine to have a class that controls capital as long as they're constrained in certain ways and pushed down certain avenues in a way that benefits the class with no capital (I've made that term up, I just wanted to give it an apparently accurate name. Normally, if it isn't labelled social democratic, it'd be named Third Way politics, or some variation in this).
It's not entirely clear whether this is true, because the differences are difficult to distinguish in practice - if you looked at the policy wishlist of a Third Wayer and a very gradual reformist socialist for their society over the next twenty years, you'd probably find absolutely minimal difference. That's why, say, I can and did support Hillary Clinton. When it comes to someone like Tony Blair, who made some pretty eloquent defences of socialism:
I am a Socialist not through reading a textbook that has caught my intellectual fancy, nor through unthinking tradition, but because I believe that, at its best, Socialism corresponds most closely to an existence that is both rational and moral. It stands for co-operation, not confrontation; for fellowship, not fear. It stands for equality, not because it wants people to be the same but because only through equality in our economic circumstances can our individuality develop properly.
which one is he? An incredibly gradual reformist socialist who recognised the political necessities of his time? Or no socialist at all?
But regardless, if you wanted a slightly faster pace than the current status quo, it became a good insult to imply that people more gradual than you actually just weren't socialists at all. Given many of these people were moving away from socialist as a term because of the increasingly negative connotations thanks to the slight of hand I mentioned earlier and embracing social democrat as a 'safe' word, social democrat became a weapon, almost. It's like neoliberal - it's not clear to me social democrat actually has much of a concrete meaning at all. Relatively few people self-identify as social democrats, and those that do perceive it totally differently to those who don't.
I actually think Sanders is a socialist. I think his ideal world is classless, that genuinely is his end-point. But he's not a revolutionary socialist, by any means.