Conditional volatility is the volatility of a random variable given (i.e. conditioning on) some extra information. E.g. in the GARCH model the conditional volatility is conditioned on past values of itself and of model errors (see below). Unconditional volatility is the "general" volatility of a random variable when there is no extra information (no conditioning).
Realized volatility is the empirical unconditional variance over a given time period. E.g. if 5-minute returns on a stock price are collected over a trading day, their empirical variance can be called realized volatility ("realized" in the sense that it has already been measured). Recall that variance is a property of the data generating process that is unobservable and can only be measure with imperfect precision from the data.
Does that mean that e.g. a Garch model can be used to model volatility or do I use Garch to model conditional volatility?
The GARCH($s,r$) model looks like this:
$$
\begin{aligned}
r_t &= \mu_t + u_t, \\
u_t &= \sigma_t \varepsilon_t, \\
\sigma_t^2 &= \omega + \sum_{i=1}^s\alpha_i u_{t-i}^2 + \sum_{j=1}^r\beta_j \sigma_{t-j}^2, \\
\varepsilon_t &\sim i.i.d.(0,1).
\end{aligned}
$$
It specifies the conditional distribution of a random variable $r_t$ conditional on past values of the model errors $u_t$, conditional variances $\sigma^2_{t-j}$ and whatever other variables that determine the conditional mean $\mu_t$.
The unconditional variance $\sigma_t$ of the error term $u_t$ is given by $\frac{\omega}{1-\sum_{i=1}^s\alpha_i u_{t-i}^2 + \sum_{j=1}^r\beta_j \sigma_{t-j}^2}$, while the unconditional variance of $r_t$ is generally more messy as it also involves $\mu_t$ which may be arbitrarily complicated. (However, the conditional mean is often taken to be as simple as $\mu_t=0$ in which case the conditional and unconditional variances of $r_t$ equal those of $u_t$.)
I have read some papers which use particular models to model conditional vola and sometimes call it conditional vola and sometimes just vola.
I think "conditional" is sometimes omitted for brevity if it is supposed to be clear from the context. Otherwise it is mentioned explicitly to avoid ambiguity.
