In MRI (Magnetic Resonance Imaging), you manipulate protons to get a signal. This is, of course, a quantum mechanical process - luckily it is possible to describe the process in a semi-classical way as you did. In this case, you do not consider individual protons, but small volume elements (voxels). In those voxels, there are enough proton spins to average over, which will give you the magnetization vector of this voxel.
If left alone in the external magnetic field $B_0$, this vector will grow along the field and reach a certain "lengths", i.e. a certain "amout" of magnetization that does depend on the strength of the external field.
If an external RF field is applied for excitation, this vector is deflected towards the transverse plane. Since there is a precession around the external field $B_0$, this vector will spin with the frequency equal to the Larmor-frequency. It is therefore convenient to transform the coordinate system so that it also rotates with this frequency. The result is a magnetization vector that - under perfect conditions - would not precess anymore in this rotating coordinate system.
On what you have asked
If the vector is in the transverse plane, there may be several reasons for it to have a phase. A phase in this case could mean an angle between the $x$-axis of the rotating coordinate system and the magnetization vector. This additional phase can be caused by several things, e.g.:
The exciting RF pulses phase. The irradiated RF pulse does also have a phase This is the angle between the $x$-axis of the rotating coordinate system and the magnetic component of the RF-pulse (called the $B_1$ field. This RF phase is important for several imaging sequences. For example in a spoiled gradient-echo sequence (common acronyms: FLASH, FFE), the RF phase is key to achieve good spoiling properties. The reason for this is that transverse magnetization cannot be "destroyed", what would be the ideal thing to happen in these sequences. Hence, there always is a little transverse magnetization that could spill over into the next data acquisition. By choosing an appropriate time-varying phase for each RF pulse, one can achieve that the remaining transverse magnetization does not add up coherently, and does therefore not build up a large, but unwanted signal. On the other hand, in a steady-state free precession sequence (acronyms fully-balanced SSFP (fbSSFP), TrueFISP, balanced FFE, or FIESTA) there is a certain 0°-180°-0°-180°-... pattern in the RF-pulses phase that ensures maximum coherent addition. For this reason, fbSSFP is the gradient-echo sequence with the highest signal in general. Note that the RF phase is not to be confused with the tip angle. Under perfect conditions, the transverse magnetization vector will exhibit a phase in the rotating frame due to the RF pulse's phase. The phase of the magnetization vector after excitation is constant in time under ideal conditions.
External variances in the $B_0$ field. Due to tissue borders or technical imperfections, the main magnetic field $B_0$ may vary in the object under infestigation. Hence, the transverse magnetization vector experiences a different Larmor-frequency that the one in the rotating frame. This means, it is (slowly) precessing in the rotating frame. This is the reason why $\phi$ does depend on $t$ in your formula.
The volume itself is not of that high importance. I assume, the $v_j$ in your formula is a "spin density", i.e. "spins per volume". Of course, voxels exhibiting less protons and therefore less signal-baring spins will contribute less to the signal than areas with high spin density.
I hope that helps!