Running stitch is one of the most basic stitches of all hand stitches. To do this stitch, you just have to run the thread across the fabric, in short even intervals. This is a most preferred stitch to do hand quilting. One advantage of this stitch is that the stitch will look the same on the back as well as on the front.
The running stitch can be used as a base to form a number of beautiful stitches which can be used for decorating your clothes. Borders of kids clothes and shawls can be decorated beautifully with these stitches. I love doing the threading of running stitches using metallic thread. It is very difficult to embroider with metallic thread. So using this thread to lace / loop over the base of embroidery thread of running stitch makes me add metallic thread to my work.
When doing this stitch in parallel rows you can work it as aligned rows or as stepped rows. Aligned rows will have running stitches one above the other. Stepped running stitch will be having running stitches in a brick laying pattern as shown below.
1.Whipped running stitch
The base of running stitch is done first. Then the needle is inserted through the stitches (Whipped stitches) without piercing the fabric.This stitch looks best when done in two colours of thread.
2. Double running stitch
This stitch is also known as Holbein Stitch, Line Stitch, Two-sided Line Stitch, Square Stitch and Spanish Stitch
This stitch is a reversible stitch. On both sides of the fabric this stitch will look the same. I love this as an outline stitch much more than a back stitch.
To do this stitch correctly first make a line of running stitches then come back and fill in the open spaces.Completed it looks like a line of back stitches. The filling stitches can be done in a contrasting colour.
3 Threaded running stitch / Laced running stitch
This stitch is made by first making a row of sunning stitch. Weave loose loops under the running stitches with same coloured thread or a contrasting colour without piercing the fabric. 
4 Interlaced running stitch.
First do a line of laced running stitch and then come back with your needle and thread and make similar lacing from the other end. This stitch will create round loops around the running stitch
5 Stepped threaded running stitch
This stitch is done by making two rows of stepped running stitches and looping another thread through the running stitches.
6 Looped running stitch.
To work this stitch, make 2 parallel rows of aligned running stitches
Start the needle from the first row first stitch, pass the needle and thread down the first stitch.Continue to the first stitch in second row ;Now bring back the needle under the first top stitch again.You will get a loop. Continue to the second stitch in the same way.
7 Seed stitch
This stitch refers to running stitch worked inside a design to fill it. Parallel rows of running stitches are used inside the design. The running stitches end up looking like seeds. This stitch can be done in a scattered manner as well.
8 Pattern Darning
Darning stitches which are basically running stitches are used in long and short versions in a consistent manner to make beautiful patterns. I can imagine an exquisite shawl border worked in this manner
Checkout the post on pattern darning for more details.
9 Checkerboard Running Stitch
This refers to running stitches made in rows of 3 or more in a pattern.
Checkout Kantha work and Sashiko embroidery tutorials for more variations of this versatile stitch.
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Nice photos and details