When I was a child, Barbie had a twist'n turn waist and bendable knees. Today - thanks to greater articulation in other fashion dolls like Momoko, Susie, Fashion Royalty, etc. - sometimes, Barbie can move her head, twist, fully bend her knees, bend her elbows and sometimes, bend her wrists. Here are some of the more pose-able Barbies:
Let's begin with the Harley Davidson Barbie #5 (2001), the third doll from the left. HD Barbie has the "gymnast" body used for sports Barbies in the 1990s. Her head can move up, down, down, back, side to side. Her elbows bend as do her knees. She has a twist waist (covered by the black teddy she wears).
The first doll from the left is the Cabaret Dancer Barbie from the Jazz Baby series (2007). The CD Barbie has the new "Pivotal" body, introduced with the Jazz Baby dolls. In addition to the articulation listed for the HD Barbie, CD Barbie's wrist bends. Her waist does not twist, but her upper torso can twist as seen by the "break" under her bra line. Like the HD Barbie, she could ride a motorcycle or a horse, but neither she nor the HD Barbie can sit with their legs completely akimbo as the ToyBiz Gandalf shown in the male yoga class photo below.
The last doll is the Alvin Ailey American Dance Theatre Barbie (2009). As you can see in the photos, she and the CD Barbie have the same bodies EXCEPT for her modified, arched feet, more in keeping for modern/ballet dance positions. The CD Barbie feet are flatter and less arched for cabaret dances.
Both the Cabaret Dancer Barbie and the Alvin Ailey American Dance Theatre Barbie have thinner arms and legs than the Harley Davidson Barbie. The joints are more discreet than with the Harley Davidson Barbie.
If you are interested in reading more about articulated fashion dolls, please read this Men With Dolls thread.