The dirac notation itself, the $|$ and $\rangle$ parts is simply a notation to remind you that you're dealing with quantum states. What you write inside this `ket' is completely arbitrary.
In the examples you give, somebody has (probably) chosen to depict a quantum system of two quantum bits. The up arrow and the down arrow depict two distinguishable states, i.e. we could measure each qubit and ask "is it up or down"? They have also made an implicit connection between an ordering of the labels and the qubits themselves.
So, the first example says that both qubits are "up". The last one says that they are both "down".
The middle two are very similar, so let's just talk about $\left|↑↓\right\rangle+\left|↓↑\right\rangle$. You could effectively read this as saying "if you ask which of the two qubits is up or down, you have a 50:50 chance of the first being up and the second down, or vice versa".
You asked for an example using superposition. Examples 2 and 3 that you gave are using a particular type of superposition called entanglement. Basically, whenever you have more than one ket, you've got superposition. When the superpositions span different physical entities (such as two qubits), that superposition is entanglement.
But we can talk about superposition of a single qubit. If we start from $\left|↑\right\rangle$, i.e. the qubit is "up", we can talk about any superposed state of the form
$$
\cos\theta \left|↑\right\rangle+\sin\theta e^{i\phi}\left|↓\right\rangle.
$$
If you asked whether it's up or down, you'd get answer "up" with probability $\cos^2\theta$, and "down" with probability $\sin^2\theta$.
The problem is if you only talk about the question "up or down?" you can't really see anything quantum happening. To do that, you have to perform measurements in other bases, and then you really start to need to do the maths properly.
