LSM303DLHC low-power and high precision modes
&quot;High precision&quot; is used as a close synonym with &quot;normal&quot; mode, and is the opposite of &quot;low power&quot; mode. From the datasheet, it appears that the high precision mode samples the raw sensor 4 times for each output sensor value.
Output Data Rate (ODR) is the fastest rate that the sensor produces data (and hence that you should read that data). Freescale has this to say (source):

Q: Once the previous measurement has been read, how long does it take to get the next one? What is the duration between two DRDY (data ready) flags? Does the 80-Hz output data rate mean 12.5 ms between two DR flags?
A: The 80-Hz output data rate means 12.5 ms between two DR flags.

So, for the LSM303DLHC you can read the accelerometer at up to 1.344 kHz in normal mode or 5.376kHz in low-power mode. Notice how 5.376kHz is 4x as fast as 1.344kHz.
Bandwidth refers to the maximum frequency signal that you can reliably measure. Analog Devices has a nice quote on the subject:

Output Data Rate, in digital-output accelerometers, defines the rate at which data is sampled. Bandwidth is the highest frequency signal that can be sampled without aliasing by the specified Output Data Rate. Per the Nyquist sampling criterion, bandwidth is half the Output Data Rate.

For the LSM303DLHC the fastest signal that the accelerometer can measure in normal (high-precision) mode is 1344Hz/9 = 149.3Hz. In reality you'll need to give yourself some margin since the signal will likely not be repeating infinitely.