Different koshinages

Main ones

Ogoshi - Basic big koshinage - nage's arm goes around uke's back/waist (back of hip versus front of hip for uki goshi)
kogoshi - nage's arm around uke's neck
Ukigoshi - koshinage off the front of nage's hip (ukashi goshi)
Tsurube goshi - Holding the sleeve koshinage; nage grabs both of uke's arms; nage's arm is straight and is grabbing uke' upper arm (as opposed to nage's arm being around uke's back) - Starting from left hanmi, shomenuchi attack - irimi to uke's left side and tenkan on your right foot; lead uke around by grabbing his left wrist; grab his right wrist with your free left hand and step in between his legs with your left foot; extend uke's right arm up and over your head and do koshinage.
kata garuma goshi: Uke holds nage's left hand. Nage steps in between uke's legs with his right foot and atemis to uke's face with his right hand. Extend your left arm over your head as you bring your left foot in next to your right foot. Lower your hips and lead uke up and over your hips with your left arm. (garuma is 'wheel', rotate like a wheel)