Actually, in simple terms, Marxism is the economic and political theory developed by Karl Marx, Friederich Engels, and all other Marxist theorists.
Original Marx's theory didn't provide a complete idea for a political system. Marx worked on a critical analysis of the economic foundations of Capitalism as it was manifested in economic developed countries in Europe, during the first half of the 19th century (basically England, as you can see on his classic economic theory work, The Capital).
That economic critique provided the basic framework for his historical analysis of the forthcoming development of human societies, and hence to his original political theory, where he outlined the idea of a society without social classes, with plenty of resources to all, without state and without private property, which he conceptualized as communism (as per The Communist Manifesto).
As Marx understood, upon all human societies achieving communism, there would be no need for a state or coercive system of law, as there wouldn't be any possible remaining disputes between people, as the reasons for common disputes, again accordingly to Marx, private property, scarcity of resources, unbalance in the distribution of scarce resources, wouldn't exist anymore, and, as result, all people would live in permanent peace, with all resources they needed being freely and immediately available to all as soon any need arose.
In order to be able to transition between the capitalist society and the bourgeois state into that ideal end of history, the communism, the labor class (the proletariat, "those that own just their own children, or prole") would have to organize itself in a party or movement to overthrow the bourgeois state and establish their own socialist state, the proletarian dictatorship, where all means of production would be socialized, so becoming property of all through the state as the representative of the working class, and the proletarian government would work to eliminate all remains of the bourgeois society, state and culture, in order to prepare the whole mankind to achieve the communism.
So, summarizing, MARXISM is the economic and political theory that provides the critique to the capitalist and bourgeois society, and from that critique develops a political revolutionary theory that defines a type of SOCIALIST state that, accordingly to Marx, would be needed to destroy the previous existing order and to build a new order, which he defines as a "perfect utopia" of a stateless and classless society with no scarcity of resources and free distribution of all those resources to all, and which he calls COMMUNISM.
That is basically, in very simple terms, the Marxist theory meanings for those three concepts you asked for.
However, bear in mind that Socialism may refer to different political currents that are not necessarily Marxist (some even predate Marx), or while built under original Marxist influences, "reformed" themselves and denied the need of an armed revolution or the socialist proletarian dictatorship.
Those "reformed", but originally Marxist-influenced parties, would become what was known as Social-Democracy, until the 1990s, as those parties changed their platforms in order to accommodate the possibility of building a socialist society (with some of them still keeping the final goal of a communist classless/stateless society in their programs) through engaging in the traditional democratic liberal process, within the bourgeois state framework, including traditional elections, building social and cultural organizations, doing propagandist actions through traditional bourgeois-controlled media, and so on.
I'll tackle this from the perspective of a political historian - others have talked about the theory aspect. Fair warning, in politics these are all pretty fuzzy terms, because socialist, communist, and Marxist are often treated as buzzwords or epithets, regardless of their actual meaning.
The short version is that SOCIALISM is a group of political theories and ideologies that emphasize equality. The early socialist thinkers (first half of the 19th century), people like Charles Fourier, Robert Owen, and Count Henri de Saint-Simon, were largely reacting to the then still-new Industrial Revolution. The early industrial era was a bad time to be a worker, with extremely long hours (14+), low wages, dangerous working conditions, and no child labor laws. The Socialists believed that this was bad, and wanted to create a more equal and just society -- how they'd do this, and what kind of society they'd make, is a point of debate among socialists, but it usually involves more active government involvement in the economy.
Karl Marx, and MARXISM is a specific socialist theory, written by Marx and Engels in the second half of the 19th century. Marxism's big idea is that the world is made up of economic classes (workers / businessmen / aristocrats, loosely), and that struggle between classes is what drives the world forward. Eventually, the workers would take over and institute an equal society. (I'm simplifying a lot here - Marx wrote a lot of very thick books).
Marxism is the biggest and most popular branch of socialism, although its not an only one. In the late 19th century, there were a lot of socialist and specifically Marxist political groups, and they had a big internal argument. Basically, one group thought that the best way of putting the workers in charge was to operate through the democratic system. Get out the vote, win elections, put in reforms, make stuff better. The second group thought that the only way to win was through violence, revolution, and military coups -- they figured that the bosses would never really let a democracy take away their stuff. Eventually, the two groups split up -- they both agreed on the goal (workers taking over + equal society), but not on the methods (reform vs. violence).
The first group, the reformers, became known as the SOCIAL DEMOCRATS, or Democratic Socialists. In a lot of places, they're the main left-wing party, either on their own, or allied with Social Liberals (different intellectual tradition, comes out of Liberalism, but shares methods but not goals with Social Democrats). In the United States, the Democratic Party could be called an alliance of Social Liberals and Social Democrats, although some are taking a more directly Democratic Socialist stance nowadays. In Europe, you've got the Labour Party in Britain, the Social Democrat Party in Germany, and the Socialist Party in France as pretty straight examples of Social Democrats. In pre-1917 Russia, these guys were called the Mensheviks.
The second group, the revolutionaries, are generally called COMMUNISTS or Marxist-Leninists, because it's Lenin, the guy who took over Russia in 1917, who really talked about this need for violence and revolution. His party, the Bolshevik Party, showed that Communists could be quite successful at taking power -- Communism is big on organization, unity, discipline, and so on -- but also that once in power, it's really easy for a communist state to slide over into dictatorship, especially as unlike Social Democrats, Communists don't like too share power with other political factions. The Soviet Union is the classic example of a Communist country, but Poland, Bulgaria, Vietnam, China... there aren't very many 'real' communist countries left these days, since the Soviet Union heavily discredited the ideology (they spend seventy years and so much energy without actually making much equality). There are countries which still call themselves Communist (China, North Korea), but they're not really -- China is pretty capitalist nowadays, North Korea is this odd nationalistic/fascistic/divine monarchical thing.
Sources
Much of this is drawn from my historiography classes (history of history, basically), particularly where it touches on Russia. I've particularly used Peter Kenez's A History of the Soviet Union, Franco Venturi's Roots of Revolution, and W. Bruce Lincoln's In War's Dark Shadow, though those are a bit beyond high school level*.* I've also read modest bits of Marx and Lenin.
Fordham University also has a bunch of original documents available online, found https://sourcebooks.fordham.edu/mod/modsbook33.asp . This is ridiculously useful and cool to read, and it's already been shaved down to the interesting or easy-to-understand bits, since it's meant for classroom use. So you can go and find Fourier and read how he writes, about fifty years before Marx, that "Liberty, unless enjoyed by all, is unreal and illusory. . .to secure liberty a Social Order is necessary which shall (1) Discover and organize a system of industry; (2) Guarantee to every individual the equivalent of their natural rights; and (3) Associate the interests of rich and poor."
Anyway, hope this helps!