Consider a vector of maturities $(m_{1},\ldots,m_{N})$ and corresponding spot (zero) rate vector at time $t\geq0$ (which could be obtained from bootstrapping for instance, or in the case of LIBOR rates already given):
$$\left(S_{t}^{(m_{1})},\ldots,S_{t}^{(m_{N})}\right).$$
This can be viewed as a stochastic process with state space $\mathbb{R}^{N}$ or as $N$ individual stochastic processes $\{S^{(m_{j})}_{t}\}_{t\geq0}$ for $m_{j}\in\{m_{1},\ldots,m_{N}\}$ with state space $\mathbb{R}$. Corresponding to this process is a forward process
$$\left\{\left(f_{t}^{(m_{1})}(T),\ldots,f_{t}^{(m_{N})}(T)\right)\right\}_{t\geq0}$$
that has an extra (deterministic) dimension in the variable $T$ and therefore the forward process has state space $\mathcal{C}(\mathbb{R})^{N}$. Let us isolate our attention to one component of this process, say $f_{t}^{(m_{j})}(T)$ for some $1\leq j\leq N$, the forward spot rate process corresponding to maturity $m_{j}$ and future time $T$. How do we determine this process in terms of $S^{m_{j}}_{t}$ so that for each $T\geq0$ the usual arbitrage conditions are satisfied? For a non-dividend paying stock, the process is given by
$$f_{t}(T)=S_{t}e^{r(T-t)}.$$
But there is no obvious analogue for interest rates. In fact, $f_{t}^{(m_{j})}(T)$ will depend on two components of the spot rate process for each $T$. The definition is
$$(1)\;\;\;\;f^{(m_{j})}_{t}(T)=\frac{S^{(T+m_{j})}_{t}(T+m_{j})-S^{(T)}_{t}T}{m_{j}}.$$
This is obtained by equating the cash flows of borrowing at $S^{(T+m_{j})}_{t}$ for $T+m_{j}$ years versus borrowing at $S^{(T)}_{t}$ for $T$ years and then rolling forward the returns at $f^{(m_{j})}_{t}(T)$ for $m_{j}$ years. This of course restricts $T$ to the set $\mathcal{T_{j}}:=\{m_{i}\}_{1\leq i\leq N-k(j)}$ where $k(j)$ is such that $T+m_{j}\leq m_{N}$ for all $T\in\mathcal{T_{j}}$.
An equivalent and more commonly encountered way to express this is to write
$$(2)\;\;\;\;f^{(m_{j})}_{t}(T_{1},T_{2})=\frac{S^{(T_{2})}_{t}T_{2}-S_{t}^{(T_{1})}T_{1}}{T_{2}-T_{1}}.$$
In this case, we require $T_{2}-T_{1}=m_{j}$ and $m_{1}\leq T_{2}\leq m_{N}.$ Of course, if we are interested in a forward rate at time $T$ that is not compatable with the spot rate vector that we have available at time $t$, then we can interpolate the pot curve (although this could potentially lead to arbitrage opportunities if you (a bank, say) are using this forward curve to price fixed income derivatives such as a swaps, because it is conceivable that a potential arbitrageur could get a spot rate for that interpolated term date different from your interpolated value!).
If we compute (1) for all $T\in\mathcal{T}_{j}$ we get a vector of points that represents the forward curve for the spot rate $S^{(m_{j})}_{t}$ associated with term $m_{j}$ computed at time $t$. If we then do this for each $1\leq j\leq N$, we get a set of forward curves, one for each spot rate term. If we fix a $T$ and plot $f^{(m_{j})}_{t}(T)$ for each $m_{j}$, then we get a forward term structure curve corresponding to the forward time $T$. The particular interpretation taken depends on the context; usually the former is used for evaluating swap derivatives and the ladder for evaluating bond derivatives.
