To start, that site Diffen is pretty terrible. I honestly think you'd do better reading a Wiki on each of those schools of thought and doing your own comparison.
The names of the "Democratic Party" and the "Republican Party" have little to do, in today's world, with the concepts of democracy and republic. You can read
http://en.wikipedia.org/wiki/Republican_Party_(United_States)#Name_and_symbols and
http://en.wikipedia.org/wiki/Democratic_Party_(United_States)#Name_and_symbols to compare.
Party affiliation/political leaning doesn't always neatly align with positions on certain issues, and oftentimes, not agreeing on one issue doesn't mean you aren't "conservative"/"liberal"/whatever. Take gun rights, for instance. There is a whole history of U.S. settler colonialism and racism behind the American emphasis on gun rights (and the politics of who it is extended to). American conservatives tend to favor more gun rights and "liberals" tend to favor fewer. Yet, historically, leftists have upheld the working class' right to bear arms.
When thinking about politics, I urge you to not limit your understanding to the U.S. Why? Because the U.S. is an anomaly in the world with its political terminology, long history of a two-party system, and the ideological and practical closeness of the two dominant parties. Not to mention, the lack of substance in electoral politics. Other countries have lively political debates and processes, with a range of contending political parties. So "conservative" and "liberal" is a false dichotomy that is a product of the basically empty political life in the U.S., regardless of how their hardcore ideologues try to act like their slight differences make them polar opposites.
Federalist/Anti-Federalist, in those terms, had more relevance in the early years of the U.S. Federalists basically favored a strong federal government and anti-Federalists opposed it. It really isn't used in the present-day, but obviously the spirit of that debate lives on.
Communism and fascism are completely different, and it is only through misinformation and smears that the two can even be used in the same breath without talking about how diametrically opposed they are. First, communism describes a stateless, classless society, predicated upon conditions of material abundance (summed up in, "from each according to his ability, to each according to his need"). Socialism, a system under which workers operate and control the means of production and goods are produced for people, not profit, would eventually evolve into communism once the productive forces of society allowed. This is the
actual left when it comes to politics. Fascism is a right-wing ideology that basically fuses business and the state. A core principle is class collaboration, which holds that the nation-state is the foremost identity and institution in society, and people - regardless of whether one is a member of the bourgeoisie or the working class - all have their role to play in society and shouldn't question their positions. Fascism typically comes on the heels of a failed working class revolution, and represents the purest form of capitalism, when the capitalists/bourgeoisie are compelled to resort to the naked force of the state to maintain their rule.
On a side note, CNN and mainstream media is garbage breh. I can recommend some news sources if you're interested. We talk about politics often on here so I could also link you to some threads that may help you get a better understanding of some of these things.