(IR FGR) A gradient echo sequence with an inversion pulse.

See Gradient Recalled Echo Sequence, Inversion Recovery and Steady State Free Precession.

(MPGR) Multiplanar gradient recalled acquisition in the steady state is a term for a fast gradient echo sequence with slice selective RF pulses.

See Gradient Recalled Echo Sequence and Gradient Echo Sequence.

(GRE - sequence) A gradient echo is generated by using a pair of bipolar gradient pulses. In the pulse sequence timing diagram, the basic gradient echo sequence is illustrated. There is no refocusing 180° pulse and the data are sampled during a gradient echo, which is achieved by dephasing the spins with a negatively pulsed gradient before they are rephased by an opposite gradient with opposite polarity to generate the echo.
See also the Pulse Sequence Timing Diagram. There you will find a description of the components.
The excitation pulse is termed the alpha pulse α. It tilts the magnetization by a flip angle α, which is typically between 0° and 90°. With a small flip angle there is a reduction in the value of transverse magnetization that will affect subsequent RF pulses. The flip angle can also be slowly increased during data acquisition (variable flip angle: tilt optimized nonsaturation excitation). The data are not acquired in a steady state, where z-magnetization recovery and destruction by ad-pulses are balanced. However, the z-magnetization is used up by tilting a little more of the remaining z-magnetization into the xy-plane for each acquired imaging line.
Gradient echo imaging is typically accomplished by examining the FID, whereas the read gradient is turned on for localization of the signal in the readout direction. T2* is the characteristic decay time constant associated with the FID. The contrast and signal generated by a gradient echo depend on the size of the longitudinal magnetization and the flip angle. When α = 90° the sequence is identical to the so-called partial saturation or saturation recovery pulse sequence. In standard GRE imaging, this basic pulse sequence is repeated as many times as image lines have to be acquired. Additional gradients or radio frequency pulses are introduced with the aim to spoil to refocus the xy-magnetization at the moment when the spin system is subject to the next α pulse.
As a result of the short repetition time, the z-magnetization cannot fully recover and after a few initial α pulses there is an equilibrium established between z-magnetization recovery and z-magnetization reduction due to the α pulses.
Gradient echoes have a lower SAR, are more sensitive to field inhomogeneities and have a reduced crosstalk, so that a small or no slice gap can be used. In or out of phase imaging depending on the selected TE (and field strength of the magnet) is possible. As the flip angle is decreased, T1 weighting can be maintained by reducing the TR. T2* weighting can be minimized by keeping the TE as short as possible, but pure T2 weighting is not possible. By using a reduced flip angle, some of the magnetization value remains longitudinal (less time needed to achieve full recovery) and for a certain T1 and TR, there exist one flip angle that will give the most signal, known as the "Ernst angle".
Contrast values:
PD weighted: Small flip angle (no T1), long TR (no T1) and short TE (no T2*)
T1 weighted: Large flip angle (70°), short TR (less than 50ms) and short TE
T2* weighted: Small flip angle, some longer TR (100 ms) and long TE (20 ms)

Classification of GRE sequences can be made into four categories:

• T1 weighted or incoherent/(RF or gradient) spoiled GRE sequences
• T1/T2* weighted or coherent/refocused GRE sequences
• T2 weighted contrast enhanced GRE sequences
• ultrafast GRE sequences

See also Gradient Recalled Echo Sequence, Spoiled Gradient Echo Sequence, Refocused Gradient Echo Sequence, Ultrafast Gradient Echo Sequence.

In fast imaging sequences driven equilibrium sensitizes the sequence to variations in T2. This MRI technique turns transverse magnetization Mxy to the longitudinal axis using a pulse rather than waiting for T1 relaxation.
The first two pulses form a spin echo and, at the peak of the echo, a second 90° pulse returns the magnetization to the z-axis in preparation for a fresh sequence. In the absence of T2 relaxation, then all the magnetization can be returned to the z-axis. Otherwise, T2 signal loss during the sequence will reduce the final z-magnetization.
The advantage of this sequence type is, that both longitudinal and also transverse magnetization are back to equilibrium in a shorter amount of time. Therefore, contrast and signal can be increased while using a shorter TR. This pulse type can be applied to other sequences like FSE, GE or IR.

Sequences with driven equilibrium:
Driven Equilibrium Fast Gradient Recalled acquisition in the steady state - DE FGR,
Driven Equilibrium Fourier Transformation - DEFT,
Driven Equilibrium magnetization preparation - DE prep,
Driven Equilibrium Fast Spin Echo - DE FSE.

Spoiled gradient echo sequences use a spoiler gradient on the slice select axis during the end module to destroy any remaining transverse magnetization after the readout gradient, which is the case for short repetition times.
As a result, only z-magnetization remains during a subsequent excitation. This types of sequences use semi-random changes in the phase of radio frequency pulses to produce a spatially independent phase shift.
Companies use different acronyms to describe certain techniques.

Different terms for these gradient echo pulse sequences:
CE-FFE-T1 Contrast Enhanced Fast Field Echo with T1 Weighting,
GFE Gradient Field Echo,
FLASH Fast Low Angle Shot,
PS Partial Saturation,
RF spoiled FAST RF Spoiled Fourier Acquired Steady State Technique,
RSSARGE Radio Frequency Spoiled Steady State Acquisition Rewound Gradient Echo
S-GRE Spoiled Gradient Echo,
SHORT Short Repetition Techniques,
SPGR Spoiled Gradient Recalled (spoiled GRASS),
STAGE T1W T1 weighted Small Tip Angle Gradient Echo,
T1-FAST T1 weighted Fourier Acquired Steady State Technique,
T1-FFE T1 weighted Fast Field Echo.
In this context, 'contrast enhanced' refers to the pulse sequence, it does not mean enhancement with a contrast agent.