It is the basic standing asana in most forms of yoga with feet together and hands at the sides of the body. There is some contention between different styles of yoga regarding the details of the asana which results in some variations.
Although Tāḍāsana is a very basic asana, it is the basis and starting point for many standing asanas.As such, Tāḍāsana is important in the context of other āsanas because it allows the body and consciousness to integrate the experience the preceding āsana and prepare for the next.  Further, as a common and fundamental asana, Tāḍāsana can help develop habits for further practice and asanas.
Due to terminological differences, Samasthitiḥ is sometimes considered a variation of Tāḍāsana.
Placing the feet wider is common in vinyasa styles of yoga and provides a more stable base in this and other such standing asanas.
A variation called Prayer Pose involves the hands resting at the "heart's center" (in front of the sternum) in Namaste.
Further variations include turning the feet outwards so they point to the sides while keeping the heels together, with the hands together at the chest (in Anjali mudra), or palms pressed together behind the back in Paścime Namaskāra (Reverse Prayer Pose)
There are also asanas which stem from Tāḍāsana but are not standing asanas, such as Supta Tāḍāsana (Reclined Mountain Pose) 
In Iyengar yoga, when performing Tāḍāsana, the arms can be raised over the head or kept at the sides of the legs. In Iyengar yoga styles, Samasthitiḥ is synonymous with Tāḍāsana. 
Tāḍāsana appears in the 1st and 2nd weeks, of Iyengar's āsana courses (as detailed in Light on Yoga).
Vinyasa Krama Yoga
In the Vinyasa Krama Yoga Tāḍāsana is performed on the toes, while samasthitiḥ is flat footed. In this style of yoga the two āsanas are different.
In Vinyasa Krama Yoga, Samasthitiḥ is the centerpiece of the standing sequence and the foundation for the Hasta Vinyasas (arm vinyasas), Parsva Bhangi (side) vinyasas, Uttanasana (forward bending) vinyasas, and squatting/hip stretching asanas.
In the Vinyasa Krama Yoga standing sequence, the final asana of the series (before śavāsana) is Tāḍāsana, performed on the toes.
In Ashtanga Vinyasa Yoga, Tāḍāsana is the beginning and ending asana in the Sūrya Namaskāra sequence which is used to warm up the body and is sometimes interspersed throughout the entirety of all Ashtanga Series if performed with full vinayasas, in addition to being a foundational pose for all standing poses.
Key aspects of Tāḍāsana are:
Feet are together or if apart, then the heels and big toes are in one line pointing straight forward.
Feet are stretched on the floor, including metatarsals and each individual toe.
Feet "ground" primarily through the three points which make up the foot's three arches: the calcaneal tuberosity (heel), the base of the first metatarsal(big toe "knuckle"), and the base of the fifth metatarsal (little toe "knuckle").  The weight is not placed on the inside edge of the feet (which would negatively affect posture throughout the body.)
Knees are tightened (in effect, the knee caps pulled up) and facing forward.
Keeping the balls of the feet grounded, and activating muscles to separate the soles of the feet (which are not allowed to move because of fricition with the mat), may help internally rotate the thighs, turn the femurs inward and keep the kneecaps facing forward. 
Hips and buttocks (upper thigh muscles) are contracted (upwards). 
Weight is distributed evenly between heels and toes, with the pelvis centred and body in-line. The pelvis is centred when the tendons which connect the sartorius and rectus femoris muscles to the pelvis is half way between being pulled tight, and relaxed. 
Weight is distributed evenly between the two hips (equally between the two legs/feet), making the left and right hips level with each other.
The shoulder is rolled back (with the head of the humerus centred in the joint), and shoulder blades pulled down the back and inwards (not "winging, but not "pinched" together). The shoulders may be "looped" (lifted slightly, then brought back and down) to achieve this.
As with most yoga asana, when entering a asana, distal body parts are best arranged first, progressively adjusting inwards and putting proximal body parts in place last.
The asana strengthens the abdomen and the legs. It may help relieve sciatica, reduce flat feet, and help plantar fasciitis and heel spurs by improving the strength of deeper foot muscles which support your foot, and reducing the load on the less suited plantar fascia. 
Due to the effects of standing, prolonged practice of Tadasana and its standing variations should be done with caution by those who suffer from headaches, insomnia, or low blood pressure.
It is advised to maintain a degree of muscle tension in this asana. Moderate tension in the thighs and hips can help develop positive habits and can increase the musculature in those areas which can help prevent injury. Additionally maintaining a degree of tension requires awareness, which can help prevent injury itself. Though in this asana it is unlikely to injure one's self, this habit can be beneficial throughout yoga practice.
Hyper extension of the knee can compresses the knee joint and can strain the medial meniscus, causing knee problems.
Since the feet serve as a foundation for the rest of the body in this and other standing asanas, the position of the feet is of high importance. Minor alteration in the position of the feet affects posture throughout the body. 
The position of the body can be described anatomically as:
The ankle, hip, shoulder, and wrist joints are neutral, halfway between flexion and extension, with the elbow joints extended and forearms pronated.This means the feet are flat and parallel, and the hands are next to the thighs with the thumbs facing forward  (in a distal position).
The following points through the body are lifted upward: the arches of the feet, the pelvic floor, the lower abdomen, the rib cage, cervical spine, and the top of the head. In this position, akin to standing "smartly", the action of standing erect should do most of the lifting without much special effort from the subject. 
Downward release exists in the following parts of the body: the shoulder blades (supported by the rib cage), the tailbone, and in the foot at the heel (Calcaneus), and the first and fifth metatarsals (the three of which serve as primary contact points with the ground). The shoulders should not be thrown back, but simply relaxed neutrally.
If the hands are raised over the head, palms together, (as per the variation) this stretches the thoracic cavity and rib structures including the intercostal muscles and expands the thoracic cavity increasing lung capacity. A slight back bend could be included to accentuate the effect and stretch the front of the body, chest and ribs more.