Also known as a straight stitch, the running stitch is one of the most basic workhorses you can have in your stitch arsenal.
One great thing about the running stitch is that there's an easy way to speed it up! Normally, you would push the needle through the fabric, then pull it and the thread all the way through. However, through much practice and dealing with looming deadlines, I've discovered that you can take a longer needle and push the head through the fabric multiple times before pulling the thread through. The fabric can be scrunched up near the bottom of the needle to get more stitches in. Then you just have to pull it all the way through, being careful not to let the needle knot up.
This stitch is good for holding fabric together. The fabric around the seam will fray if left alone, but I find this stitch is often good enough to last without any further work.
This stitch will help you keep your edges from fraying. It can also be used as a decorative outer stitch if you like.
Basically, put your needle through the fabric near the edge of what you're sewing. Pull the thread all the way through. Bring the needle up to the same side of the fabric you put the needle into before. Repeat.
For more sturdy fabric, I recommend doing a running stitch, then doing a blanket stitch over it.
Whip/blanket stitchThe whip or blanket stitch is similar to the overcast stitch.
Put the needle through the fabric as with the overcast stitch. Bring it around to the front and put it down the same hole. Pull the thread almost all the way through. Bring the needle back around and pull it through the remaining loop of thread. Repeat.
I have often used this with the running stitch to 'serge' the edges. Having had access to a serger for a few months, I can say that the machine is very fast and convenient, but the hand stitches in many ways are sturdier.
Slip-stitchThis is a fantastic invisible stitch for finishing objects. I only learned this one a few months ago, because I used to just use a whip stitch around the outer edges of my projects. Using this stitch has made my work look much more professional.
This stitch is a little more complicated to explain, but is very similar to the running stitch once you have it set up.
Fold the edges of two pieces of fabric over, then pin them together with the edges inward. Stick the needle into the fabric in the valley of the seam, then push it out through the top of the folded over edge of one piece of fabric. Put the needle in the top edge of the other piece of fabric, and push it out a little further along the fabric. Put the needle in the top of the other piece of fabric and repeat. Continue this process, alternating sides each time.
When it's time to make the knot, try to put the knot below the seam. It's a little difficult and takes experimentation, but is well worth it if you can learn how to do it.