Yarn over increase
The simplest increase is the yarn-over increase which is simply worked by moving the yarn under the right hand needle then over it towards the back of the work.

The yarn over increase is intentionally obvious – it makes a big hole in the knitting which is the basis of most lace knitting.

Make one increase
The "make one" increase is also a very easy increase. Basically a make one increase is a yarn over with a twist, which effectively closes up the big hole that is made with a yarn over. Make one (usually abbreviated M1 or m1) can be made two different ways:

One way to make a M1 is to pick up the horizontal strand between two stitches and twist it.

This can be done so that it slants right by inserting the left-hand needle from back to front into strand between two stitches…

and knit normally:

or so that it slants left by inserting the left-hand needle from front to back into strand …

and knit it through the back loop:

The result of right slanting and left slanting on either side of a center stitch looks like this:

Another way to make a M1 is the Elizabeth Zimmermann method. It produces the same stitch, only in the current row rather than the row below.

Slanting right by making a loop on the right hand needle like this:

Slanting left by making a loop on the right hand needle like this:

The finished result looks quite a bit like the first version:

These increases do not – of course – have to be paired, you can use them individually if you like, or the pattern does not call for paired increases.

Bar increase
Yet another way to increase is the knit through the front and the back (kfb) increase – also known as a “bar increase”.

First knit through the front:

then – without removing the stitch from the left hand needle – knit it through the back loop:

This increase is very sturdy and does not leave any visible hole, but does make a purl bump on the knit side of the work [what some people call the "bar"].

Raised increase
One of the most invisible increases is the raised or lifted increase.

For a right leaning increase, lift the stitch below the stitch on the left needle

and knit it.

For a left leaning increase, lift the stitch below the stitch you just knit on the right needle
and knit it.

If you find that either of these leaves more of a hole than you’d like, feel free to knit them through the back loop to tighten them up.

These are the increases that Cat Bordhi calls LLinc and LRinc and demonstrates wonderfully in this video:

I love the raised increase for toe up socks:

References:

• Vogue Knitting, the ultimate Knitting Book
• Knitting Without Tears by Elizabeth Zimmermann
• The Knitter’s Companion by Vicki Square