Every tissue in the human body has its own T1 and T2 value. This term is used to indicate an image where most of the contrast between tissues is due to differences in the T1 value.
This term may be misleading in that the potentially important effects of tissue density differences and the range of tissue T1 values are ignored.
If the machine parameters are chosen, so that TR less than T1 (typically under 500 ms) and TE less than T2 (typically under 30 ms), a power series expansion of the exponential functions and then neglecting second and higher order terms yields
Mxy = Mxy0 TR/T1
thus the expression becomes independent of T2 and yields the condition for T1 weighting. Therefore a T1 contrast is approached by imaging with a short TR, compared to the longest tissue T1 of interest and short TE, compared to tissue T2 (to reduce T2 contributions to image contrast). Due to the wide range of T1 and T2 and tissue density values that can be found in the body, an image that is T1 weighted for some tissues may not be so for others.
Lesions with short T1 are (bright in T1 weighted sequences):
fat (lipoma, dermoid)
sub-acute haemorrhage (metHb)
paramagnetic agent (Gd, pituitary)
protein-containing fluid (colloid cyst)
metastatic melanoma (melanotic).

The basis of T1 weighted imaging is the longitudinal relaxation. A T1 weighted magnetic resonance image is created typically by using short TE and TR times.
The final image is a reflection of more than one of these pulse sequence parameters, weighted according to the type of sequence and its timing. T1 signals determine predominantly the contrast and brightness in this type of images but proton density will always contribute to the image intensity. The T1 dependence is mainly determined by the repetition time or any pre-pulses (such as in an inversion recovery pulse sequence).
Due to the larger longitudinal and transverse magnetization, fat has a higher signal and will appear bright on a T1 contrast MR image. Conversely, water has less longitudinal magnetization prior to a RF pulse, therefore less transverse magnetization after a RF pulse yielding low signal appearing dark on a T1 contrast image. Often, a paramagnetic contrast agent, a gadolinium compound, is administered, and both pre-contrast T1 weighted images and post-contrast T1 weighted images are obtained.

(STAGE T1W) A RF spoiled T1 weighted gradient echo sequence.
See Spoiled Gradient Echo Sequence.

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.

(FSE) In the pulse sequence timing diagram, a fast spin echo sequence with an echo train length of 3 is illustrated. This sequence is characterized by a series of rapidly applied 180° rephasing pulses and multiple echoes, changing the phase encoding gradient for each echo.
The echo time TE may vary from echo to echo in the echo train. The echoes in the center of the K-space (in the case of linear k-space acquisition) mainly produce the type of image contrast, whereas the periphery of K-space determines the spatial resolution. For example, in the middle of K-space the late echoes of T2 weighted images are encoded. T1 or PD contrast is produced from the early echoes.
The benefit of this technique is that the scan duration with, e.g. a turbo spin echo turbo factor / echo train length of 9, is one ninth of the time. In T1 weighted and proton density weighted sequences, there is a limit to how large the ETL can be (e.g. a usual ETL for T1 weighted images is between 3 and 7). The use of large echo train lengths with short TE results in blurring and loss of contrast. For this reason, T2 weighted imaging profits most from this technique.
In T2 weighted FSE images, both water and fat are hyperintense. This is because the succession of 180° RF pulses reduces the spin spin interactions in fat and increases its T2 decay time. Fast spin echo (FSE) sequences have replaced conventional T2 weighted spin echo sequences for most clinical applications. Fast spin echo allows reduced acquisition times and enables T2 weighted breath hold imaging, e.g. for applications in the upper abdomen.
In case of the acquisition of 2 echoes this type of a sequence is named double fast spin echo / dual echo sequence, the first echo is usually density and the second echo is T2 weighted image. Fast spin echo images are more T2 weighted, which makes it difficult to obtain true proton density weighted images. For dual echo imaging with density weighting, the TR should be kept between 2000 - 2400 msec with a short ETL (e.g., 4).
Other terms for this technique are:
Turbo Spin Echo
Rapid Imaging Spin Echo,
Rapid Spin Echo,
Rapid Acquisition Spin Echo,
Rapid Acquisition with Refocused Echoes