Anibadh singing is a way of singing free of any structure. It was sung in ancient times. It did not have Sur and Taal.
Nibadh singing is one with structural elements like Sur, Taal and Padh. This was formed in modern times.
Nibadh singing is of three types Prabandh, Rupak and Vastu. Out of these three Prabandh is most famous. Any composition having Taal, Sur and Shabdh (words) is known as Prabandh.
Nowadays, what we call as bandish (song) is a word derived from the word Prabandh. Words in Prabandh are known as Geet.
In old times Prabandh singing had 4 sections. Udgrah – Introduction, Melapaka – Bridge between Udgrah and Dhruva, Dhruva – Main body of the song and Abhog – The conclusion.
Nowadays, Prabandh or Bandish has 4 sections i.e. Sthayi – the beginning, Antara – the intermediate section, Sanchari – the bridge between Antara and and Abhog – the conclusion.
For more details on Nibadh and Anibadh singing please refer this post on Music of India by Sreenivasa Rao. It has very detailed info on the subject.