Disclaimer: Since you haven't stated the source from where you got those terms, I will mention the most obvious definitions for those, which occur to me - simply looking at the names.
"Qunit" refers to any quantum system whose state lies in a complex vector space whose dimension is any natural number $n$. Example of qunits are qubits, for which $n=2$ and qutrits for which $n=3$. Also check this for other less commonly used terms: Do any specific types of qudits other than qubits and qutrits have a name?
Next, remember that a system of two qubits will lie in a $2\times 2$-dimensional vector space i.e. $\Bbb C^2\times \Bbb C^2$. A system of three qubits will lie in a $2\times 2 \times 2$-dimensional vector space $\Bbb C^2\times \Bbb C^2\times \Bbb C^2$ and so on. By "$n$-dimensional" e-bits they're referring to the dimension of the vector space again. As for the "e-bit" part, that's simple. That is, the state of the system of qubits is basically not separable into individual qubit states lying in $\Bbb C^2$ each (i.e. the qubits are entangled). Check out the answers in: How do I show that a two-qubit state is an entangled state?