Hasta Mudra (Hand Gestures) for nritta

Mudras are basically of two types -
Asamyukta Hasta (or Asamyuta) (Single Hand Gestures)
Samyukta Hasta (or Samyuta) (Double Hand Gestures)
The Natyashastra mentions 28 Mudras. There are four new Mudras added to this list. These Hand Gestures are a Part of Angika Abhinaya. I shall be explaining each of these Mudras with reference to the shlokas of Abhinayadarpana.
There are 4 new gestures added to this list - Kataka, Vyagraha, Ardhasuchi and Palli

Each of these gestures can be used for a variety of purposes in the dance. The uses of each Hasta are mentioned in a "Shlok" or "Hymn" and we call in "Viniyoga".

Sr. Name – English Name –  Meaning
 1 Pataaka :                Flag
 2 Tripataaka :            3 Parts of the Flag
 3 Ardhapataaka :       Half Flag
 4 Kartarimukha :        Scissors
 5 Mayura     :             Muh-yoo-ra Peacock
 6 Ardhachandra :       Half Moon
 7 Araala :                  Peak of mountain * (to verify)
 8 Shukatunda :          Parrot’s Head
 9 Mushthi :                Fist
 10 Shikhara :             Mountain Peak
 11 Kapitta :               Goodness of wealth * (to verify)
 12 Katakaamukha :   Crab
 13 Suchi :                  Needle
 14 Chandrakalaa :     Waxing Moon
 15 Padmakosha :       Lotus Bud
 16 Sarpashirsha :       Snake’s Hood
 17 Mrigashirsha :       Deer’s Head
 18 Simhamukha :       Lion’s Face
 19 Kaangula :            Water Lily
 20 Alapadma :           Flowering Lotus
 21 Chatura :              Clever
 22 Bhramara :            Bee
 23 Hamsaasya :         Head of Swan
 24 Hansapakshika :   Wing of Swan
 25 Sandamsha :        Tongs
 26 Mukula :              Flower Bud
 27 Tamrachuda :       Cock
 28 Trishula :              Trident
 29 Ardhasuchi :         Half Needle
 30 Vyaagraha :          Tiger
 31 Palli :                    Knot
 32 Katakaa :             Retsing place

Natyasaastra

Natyasaastra is the main sourse of Nritta-Nritya-Natya-Nadanam. It is an ancient Indian treatise on the performing arts, encompassing theatre, dance and music. It was written during the period between 200 BCE and 200 CE in and is traditionally attributed to the Sage Bharata. According to the Natya Shastra by Bharata Muni; classical dance has three main aspects namely Nritta, Nritya, and Natya. I call them the 3Ns of Bharatanatyam :-)

Nritta
Nritta is the demonstration of rhythm through graceful body movements. The synchronization between rhythm and time is of utmost importance in a Nritta piece. Beat and tempo are its two vital constituents. There is no poetic meaning to emote. Its purpose is to showcase geometric patterns and complex rhythmic variations. Hand gestures, dance steps, patterns and concluding dance movements (known as Theer-maa-num) form the backbone of Nritta.
In Natyashatra it is explained that "the sheer beauty of Nritta has a great popular appeal".

Nritya; Depicting a story with facial expressions & body movements
Nritya can be described as an explanatory dance, where the lyrics of the song are conveyed using hand gestures (hasta mudras) and facial expressions (abhinaya). The dancer’s internal emotions (bhaava) should be performed in a way that should be felt by the audience. This is the reason why it is important to understand the lyrics of any song. Unless the emotions are felt inside, it is difficult to perform a flawless piece.
.... in short, Nritya = Nritta + emotions.

Natya; combination of Nritta & Nritya
Natya is achieving a story through speech, music, Nritta and Nritya. Natya is an excellent element of classical dancing through which a dancer can express every known human emotion and feeling. Natya is the dramatic element of classical Indian dance. The dancers take on the roles of characters and perform a dramatic story through music and dance.

Bharata describes in Natya Satra  that, 15 types of drama ranging from one to ten acts. The principles for stage design are laid down in some detail. Individual chapters deal with aspects such as makeup, costume, acting, directing, etc. A large section deals with meanings conveyed by the performance (bhavas) get particular emphasis, leading to a broad theory of aesthetics (rasas).
Four kinds of abhinaya (acting, or histrionics) are described – that by body part motions (angika), that by speech (vAchika), that by costumes and makeup (AhArya), and the highest mode, by means of internal emotions, expressed through minute movements of the lips, eyebrows, ear, etc. (sAttvika).