Let's consider two equivalent probability measures $\textbf{P}$ and $\textbf{P}^\theta$ on the measurable space $(\Omega,\cal{F})$. According to Radon-Nikodym's theorem, we know that there exists a martingale $\zeta^\theta$ such that \[ \zeta^\theta(t)=\mathbf{E}^\mathbf{P}_t\left(\frac{d\mathbf{P}^\theta}{d\mathbf{P}}\right) \] where $\frac{d\mathbf{P}^\theta}{d\mathbf{P}}$ is the Radon-Nikodym derivative linking the two probability measures. Let's now suppose that $\zeta^\theta$ satisfies the SDE \[ \frac{d\zeta^\theta}{\zeta^\theta}=-\theta(t)dW(t) \] This equation can be easily solved using Ito's lemma, yielding \[ \zeta^\theta = \text{exp}\left[-\int_0^t \theta_s dW_s -\frac{1}{2}\int_0^t \theta^2_s ds \right] \] Now, this is not necessarily a martingale. A sufficient (however difficult to verify) condition is the Novikov condition. For sake of simplicity, we'll assume that $\zeta^\theta$ is indeed a martingale. Under the above assumptions, if $W(t)$ is a Brownian motion under $\textbf{P}$, then the process \[ dW^\theta_t = dW_t + \theta_t dt \label{a}\tag{1} \] is a martingale under $\textbf{P}^\theta$.
A simple exercise
Let's suppose we have an economy with an asset whose dynamics in the physical measure is
\[
dX_t=\mu X_t dt + \sigma X_t dW_t
\label{b}\tag{2}
\]
and a bank account $B_t$ with the usual dynamics $dB_t/B_t = r dt$. What is the dynamics of $X$ under the risk-neutral measure?
We can start by reminding that under the risk-neutral measure, the discounted asset price $X_t/B_t$ is a martingale. Let's find the dynamics of this process under the physical measure. Since \[ d\left(\frac{X_t}{B_t}\right)=\frac{dX_t}{B_t}-\frac{X_t dB_t}{B_t^2} \] we can write \[ d\left(\frac{X_t}{B_t}\right)/\frac{X_t}{B_t}=\frac{dX_t}{X_t}-\frac{dB_t}{B_t}=(\mu-r)dt+\sigma dW_t \] If we change measure to $\textbf{P}^\theta$, we get, according to Girsanov's theorem \[ d\left(\frac{X_t}{B_t}\right)/\frac{X_t}{B_t}=(\mu-r-\sigma \theta)dt+\sigma dW_t^\theta \] We need now to choose the market price of risk $\theta$ in order to remove the drift of the process. We easily get \[ \theta = \frac{\mu-r}{\sigma} \] With this choice, the normalized asset price is a martingale, therefore this is the risk-neutral measure. In order to find out what is the dynamics of the asset price (non normalised) we simply substitute \ref{a} in \ref{b} yielding \[ \frac{dX_t}{X_t}=r dt + \sigma dW_t^\theta \] which is the known result stating that asset prices grow at the risk-free rate in the risk-neutral world.
