Let $r(s)$ be the process of a short rate. Then, by risk neutral pricing, $$ P(t,T) = \mathbb{E}^\mathbb{Q}\left[ \exp\left( -\int_t^T r(s)\mathrm{d}s\right) \bigg| \mathcal{F}_t\right].$$ Thus, the zero-coupon bond is determined completely by the short rate process. Here, $P(t,T)$ denotes the time $t$ price of a zero-coupon bond maturing at time $T$. You just take the risk-neutral expectation of the discounted payoff. The payoff is $1$ for all states of the world $\omega$ (assuming no default risk). Thus, you get the expectation of the discount factor. The risk-neutral measure $\mathbb{Q}$ uses a bank account as numeraire with $\mathrm{d}B_t=r(t)B_t\mathrm{d}t$.

Short rate models (such as Vasicek, Hull-White, CIR, etc.) specify a stochastic model for $r(s)$, typically a (perhaps multidimensional) SDE and then, you can find sometimes analytical) prices for bonds, bond options, swaptions etc.

Let $r(s)$ be the process of a short rate. Then, by risk neutral pricing, $$ P(t,T) = \mathbb{E}^\mathbb{Q}\left[ \exp\left( -\int_t^T r(s)\mathrm{d}s\right) \Bigg| \mathcal{F}_t\right].$$ Thus, the zero-coupon bond is determined completely by the short rate process. Here, $P(t,T)$ denotes the time $t$ price of a zero-coupon bond maturing at time $T$. You just take the risk-neutral expectation of the discounted payoff. The payoff is $1$ for almost all states of the world $\omega\in\Omega$ (assuming no default risk). Thus, the price of the bond is the conditional expectation of the discount factor. The risk-neutral measure $\mathbb{Q}$ uses a bank account $(B_t)$ as numeraire with $\mathrm{d}B_t=r(t)B_t\mathrm{d}t$.

Short rate models (such as Vasicek, Hull-White, CIR, etc.) specify a stochastic model for $r(s)$, typically a (perhaps multidimensional) SDE and then, you can find (sometimes analytical) prices for bonds, bond options, swaptions etc.

The easiest case is a deterministic and constant short rate $r(s)\equiv r$. Then, $$P(t,T)=e^{-r(T-t)}$$ and clearly the short rate $r$ gives you the bond price.