Kesshu gamae: heels together, toes pointed out at 45 degrees, hands folded in front of the belt (right hand clasped around left thumb). This is the default stance – when you are listening to instructions, or not doing anything specific, this is the correct way to stand.
Gassho rei: heels together, toes pointed out at 45 degrees, palms together to salute. This stance is used when entering or leaving the dojo, and to salute your Sensei and training partners.
Kaisoku chudan gamae / Byakurenken chudan gamae: feet shoulder-width apart, toes pointing forwards, knees slightly bent. Back straight, arms held parallel to the floor, elbows resting near ribs, hands in fists. This stance is used for practicing basic movements, the kihon.
Chudan gamae: feet shoulder-width apart in both directions (as if you were standing on diagonal corners of a square), front foot pointing forwards, back foot pointing out at 45 degrees. Front hand held forwards in a fist, arm parallel to the floor; back hand held nearer the jaw. Elbows near the ribs, weight evenly distributed, knees slightly bent. This is typically the stance used for attacking.
Ichiji gamae: the same as chudan gamae, but with the front hand held flat, fingers pointing slightly upwards, to guard the groin. This is (usually) a defensive stance, and is often used when kicking.
Hasso gamae: start from chudan or ichiji gamae, square up slightly (shoulders facing nearly straight on rather than held diagonally) and bring your weight slightly off your front foot. Bring the hands up, palms facing outwards, to guard the face, leaving the midsection relatively exposed. This is a defensive stance used to provoke an attack to chudan (mid-level).
These are the basic ways to stand facing a training partner during practice, the assumption being that one partner is in chudan gamae (usually the attacker) and one is in ichiji gamae (usually the defender).
Tai gamae: both partners have the same foot in front – usually expressed as hidari (left) tai gamae or migi (right) tai gamae.
Hiraki gamae: each partner has a different foot in front, causing them to be in a “mirror” stance.