(PCA) With this method images of the blood flow-velocity (or any other movement of tissue) are produced. The MRI signal contains both amplitude and phase information. The phase information can be used with subtraction of images with and without a velocity encoding gradient. The signal will be directly proportional to the velocity because of the relation between blood flow-velocity and signal intensity.
This is the strength of PCA, complete suppression of stationary tissue (no velocity - no signal), the direct velocity of flow is being imaged, while in TOF (Inflow) angiography, tissue with short T1 (fat or methaemoglobin) might be visualized.
The strength of the gradient determines the sensitivity to flow. It is set by setting the aliasing or encoding velocity (VENC). Unfortunately, phase sensitization can only be acquired along one axis at a time. Therefore, phase contrast angiographic techniques tend to be 4 times slower than TOF techniques with the same matrix.

See also Phase Contrast Sequence, Magnetic Resonance Angiography, Contrast Enhanced Magnetic Resonance Angiography, Flow Effects and Flow Quantification.

(3D MRA) The 3D angiography technique can be applied to focus on fast flowing (arterial) blood and to visualize small tortuous vessels. 3D TOF images are less sensitive to turbulent flow artifacts. The advantage of this approach is that the signal, acquired from the entire volume has an increased signal to noise ratio. Slices are defined by a second phase encoded axis, which divides the volume into 'partitions'. 3D TOF MRA is acquired with 3D FT slabs or multiple overlapping thin 3D FT slabs (MOTSA) depending on the coverage required and the range of flow-velocities under examination.
Such 3D techniques can provide equal spatial resolution along all three axes, i.e. be 'isotropic', or the partition thickness can be greater or less than the in plane spatial resolution in which case can be said to be 'anisotropic'. The circle of Willis, anatomy as well as its fast arterial flow, lends itself well to both 3D TOF and 2D or 3D phase contrast angiography.

Angiography means the imaging of veins and arteries. Magnetic resonance angiography (MRA) has a lower invasion than conventional angiography with catheter and X-ray contrast agent. Time of flight angiography (inflow) and phase contrast angiography works without contrast agents. Only in contrast enhanced magnetic resonance angiography is the use of contrast agents necessary, but the lack of side effects is an advantage of MRI contrast agents, just as the smaller dosage as used in X-ray angiography techniques.

See also the related poll result: 'MRI will have replaced 50% of x-ray exams by'

(CE MRA) Contrast enhanced MR angiography is based on the T1 values of blood, the surrounding tissue, and paramagnetic contrast agent.
T1-shortening contrast agents reduces the T1 value of the blood (approximately to 50 msec, shorter than that of the surrounding tissues) and allow the visualization of blood vessels, as the images are no longer dependent primarily on the inflow effect of the blood. Contrast enhanced MRA is performed with a short TR to have low signal (due to the longer T1) from the stationary tissue, short scan time to facilitate breath hold imaging, short TE to minimize T2* effects and a bolus injection of a sufficient dose of a gadolinium chelate.
Images of the region of interest are performed with 3D spoiled gradient echo pulse sequences. The enhancement is maximized by timing the contrast agent injection such that the period of maximum arterial concentration corresponds to the k-space acquisition. Different techniques are used to ensure optimal contrast of the arteries e.g., bolus timing, automatic bolus detection, bolus tracking, care bolus. A high resolution with near isotropic voxels and minimal pulsatility and misregistration artifacts should be striven for. The postprocessing with the maximum intensity projection (MIP) enables different views of the 3D data set.
Unlike conventional MRA techniques based on velocity dependent inflow or phase shift techniques, contrast enhanced MRA exploits the gadolinium induced T1-shortening effects. CE MRA reduces or eliminates most of the artifacts of time of flight angiography or phase contrast angiography. Advantages are the possibility of in plane imaging of the blood vessels, which allows to examine large parts in a short time and high resolution scans in one breath hold. CE MRA has found a wide acceptance in the clinical routine, caused by the advantages:

Diffusion weighted imaging can be performed similar to the phase contrast angiography sequence. The gradients must be increased in amplitude to depict the much slower motions of molecular diffusion in the body.
While a T1 weighted MRI pulse sequence is diffusion sensitive, a quantitative diffusion pulse sequence was introduced by Steijskal and Tanner. Its characteristic features are two strong symmetrical gradient lobes placed on either side of the 180° refocusing pulse in a spin echo sequence. These symmetrical gradient lobes have the sole purpose of enhancing dephasing of spins, thereby accelerating intravoxel incoherent motion (IVIM) signal loss.
Dephasing is proportional to the square of the time (diffusion time) during which the gradients are switched on and the strength of the applied gradient field. Therefore, the use of high field gradient systems with faster and more sensitive sequences, make diffusion weighting more feasible.
Areas in which the protons diffuse rapidly (swollen cells in early stroke, less restriction to diffusion) will show an increased signal when the echo is measured relative to areas in which diffusion is restricted. For increased accuracy of diffusion measurement and image enhancement, useful motion correction techniques such as navigator echo and other methods should be used. In addition to this, applying the b-value calculated by the strength and duration of motion probing gradients with a high rate of accuracy is very important.

See also Apparent Diffusion Coefficient, ADC Map, Lattice Index Map.