This question is about the order of the k-space lines that we acquire, sometimes called profile order. After we acquire a signal, we put it into our k-space matrix. Where in the matrix it goes is determined by the nature of the encoding gradients that used for that signal. This means that we can acquire the lines in k-space in an arbitrary order, if we want to. The standard way of filling k-space, one horizontal line after the other, from the bottom of k-space to the top, is called a linear (or sequential) profile order. If we decide to do the same thing but in the reverse order (top to bottom), we call it a reverse-linear profile order. These profile orders are such that the centre line of k-space (k0), which determines most of the contrast in the image, is the middle line acquired. The choice between linear and reverse-linear becomes important when other k-space manipulations are employed, such as partial Fourier (halfscan).
Centric k-space ordering (also known as low-high) is where we acquire the middle line of k-space (k0) first, and then work our way out towards the top and the bottom of k-space. It is sometimes called low-high, because the k-space lines dominated by low spatial frequencies are acquired first. Reverse-centric (high-low) is simply the opposite: it acquires the centre lines last.
So for a 128*128 matrix, for example:
- linear: -127, ... -2, -1, 0, +1, +2, ... +128
- reverse-linear: +128, ... +2, +1, 0, -1, -2, ... -127
- centric (low-high): 0, +1, -1, +2, -2, ... +127, -127, +128
- reverse-centric (high-low): +128, -127, +127, ... -2, +2, -1, +1, 0
Note: there are 256 lines from -127 to +128. Don't forget that "0" is a line too.
There is also a cyclic profile order, but this is rare and not often used.
Why would we change the order? It is sometimes important to set when the central lines of k-space are acquired, for signal-to-noise, timing or artefact-reduction reasons, and we can do this by changing the profile order.
Three more terms worth being aware of with respect to k-space are Cartesian coordinates, k-space trajectory and segmented k-space. When we acquire k-space in simple left-to-right lines (in any order), we are using a Cartesian / rectilinear coordinate system. This is sometimes used to differentiate between other k-space trajectories, such as spiral or radial. (You can start from any point in k-space and go in any direction with the right gradient manipulations, and radial and spiral are examples of unusual trajectories. They are advanced techniques and come with their own complications and artefacts.) Basic MRI uses only Cartesian k-space trajectories, so don't worry about others if you haven't heard of them.
Segmented k-space refers to acquiring k-space lines in groups, with interruptions between the groups. (Acquiring all lines consecutively is called a "single shot".)
Further reading on this topic:
Books: MRI From Picture to Proton p128-129, 249, 288