Make slip knot and knit first stitch on. For each new stitch after that, insert the right needle between 2 stitches; wrap yarn as for knitting(A). Draw through a new stitch and transfer it to left needle(B).

Method 1—Emily Ocker’s Circular Beginning
First, you are going to make a ring with the required number of stitches, only these are going to be crocheted.
1. Make a ring. Do not pull it tight; keep it loose. Grasp the loop where the strands meet with your left thumb and index finger.
You can make the loop with the tail going up. The tail will have to be woven in after several rounds.
2. Insert the hook into the loop from front to back. Grab the yarn with the hook (yo) and bring through the loop. You now have one loop on the hoop.
3. Insert the hook into the loop again; you will be going under the yarn tail as well. Hook the yarn (yo) and bring through the loop. Hook the yarn again and bring through the loop on the hook as if you were making a chain. You now have two loops on the hook. Repeat this step until you have the required number of loops.
4. Carefully replace the hook with your needles and begin knitting in the round. After you have finished a few rounds, pull the tail to close up the circle.
You now have a nice, tight circle with no hole in the middle.
—updated January 7, 2017

Crocheters! Check here for a similar method to begin crocheting in the round.

Method 2
Cast on the required number of stitches and divide evenly between two needles, not the usual 3 or 4 needles. Using a third needle, knit in the round for a few rounds. If the pattern includes increases, when you have 12 or more stitches, it will probably be comfortable for you to now introduce another needle. Think "Magic Loop," only with double pointed needles.
—added April 23, 2011 (first posted on sheepthrillsyarncompany.com January 2011)

Method 3
Make a slip knot, leaving a 12" tail. The knot should be snug, not tight. Hold the knot in the left hand and the needle in the right hand. *Yarn over (1 st created); insert the needle into the slip knot’s loop, wrap yarn around needle and pull through the loop (another st created). Rep from * until you have the desired number of stitches. After you have knitted 3 or 4 rounds, pull the tail of the slip knot to close up the hole.
—revised March 19, 2013 from "Knitting in the Round" by the House of White Birches, pg. 162

• Make a slip knot and place it on a crochet hook the same size as the needles. (Figure 1)
• With yarn in back of the knitting needle, bring the hook over the top of the needle (Figure 2), catch the yarn and pull it through the loop on the hook. (Figure 3) Repeat this step until the number of stitches desired has been cast on.

• Step 1: Knit the next st and leave on needle.
• Step 2: Insert the tip of the left needle into the new st and slide to left needle.

• Often, we run out of the “tail” when casting on a large number of stitches. To avoid this, use one end of the skein as your working yarn and the other end (or another ball of yarn) as the tail and cast on. Your yarn tail will never be too short again!
  —added January 17, 2012, Esther Bozak, Twisted Stitches
  
• The tail of the yarn, when using a long-tail cast on, should be about 4 times the width of your finished edge.
  —added March 16, 2013
• Wind your yarn around the needle for the required number of stitches. Add a few inches.
• Measure a small length of tail; cast on stitches until you use up the tail. With a little math, you can then determine how many inches you need.
  —added January 9, 2017

Picot cast on

• Using a cable cast-on or other knit cast-on, cast on 4 stitches.
• Now we are going to begin a 5-stitch picot. Cast on 5 more stitches. Bind off 5 stitches. (Try the modified conventional bind-off; once learned, it is faster.) Slip the stitch from the right- to the lefthand needle by inserting the lefthand needle in the stitch from front to back. You now have 4 stitches.
• *Cast on 9 stitches (4 stitches plus the 5 stitches for the picot). Bind off 5 stitches. Slip the stitch from the right- to the lefthand needle by inserting the lefthand needle in the stitch from front to back. You have just added 4 stitches to the total cast on.*
• Repeat between * * until you have 4 stitches less than the total desired.
• Cast on 4 stitches.
  —added November 8, 2011 from Lucy Neatby’s Knitting Gems

• Using a smooth yarn in a contrasting color, make a slip knot and place it on a crochet hook the same size as the needles. (Figure 1)
• With yarn in back of the knitting needle, bring the hook over the top of the needle (Figure 2), catch the yarn and pull it through the loop on the hook. (Figure 3) Repeat this step until the number of stitches desired has been cast on.
• Chain a few stitches. Cut the yarn and pull through the last chained stitch. When you are ready to unravel the waste yarn, this will be the end that you gently pull to remove the waste yarn.
• Figure 4 shows the completed cast on.
  —added December 1, 2011 from Lucy Neatby’s Knitting Gems 4
  

• Cast on half the desired number of stitches with a smooth yarn in a contrasting color the same weight as the main color.
• Purl 1 row.
• Knit 1 elongated row: *K 1, wrap the yarn around the right needle twice, repeat from * ending with K1.
• With main yarn, purl 1 row dropping the wrapped yarn.
• To add another loop of main yarn that shows through the elongated stitch, using the beginning tail of the main yarn, complete the stitch by threading it through the first stitch in the row below the needle. This is important if you need an even number of stitches.
• With main yarn, purl 1 row dropping the wrapped yarn.
• With main yarn, beginning with a knit row, work 3 rows in stockinette stitch.
• 1. Pick up the first purl bump of the main yarn, place it on the left-hand needle and knit it. (Be sure to get that first bump created when you completed the stitch in the prior step.)
  2. With yarn forward, slip the next st
  3. Repeat steps 1 and 2 to the end of the row.
  
  You have now cast on and are ready to knit the next row: *K1; with yarn forward, sl 1*, repeat between * * across the row.
  After working several rows in this manner, remove the waste yarn.
  —added January 23, 2016