(1)  Seer. Strictly the term is used for inspired seers to whom Vaidik Mantra are revealed; but loosely it is used for any inspired religious person. The inspired teacher by whom the Mantra was originally seen, or to whom it was revealed. As such Sookt of Ved has its Rishi , and so has the Mantra.

(2) In day to day language Rishi came from many sections of the society - priests, kings, commoners etc. There are seven very distinguished families of Rishi - (1) Bhaargav from Rishi Bhrigu Angiras; (2) Aangiras also from Bhrigu, who include Kanv and Bharadwaaj; (3) Aatreya from Rishi Atri; (4) Kaashyap from Rishi Kashyap; (5) Vaashishth from Rishi Vashishth; (6) Aagastya from Rishi Agastya; and (7) Kaushik from Rishi Vishwaamitra who was himself descended from King Kushik.
(Aangiras, p 89)

(3) According to Padm Puraan, 3/2, Rishi are of three kinds - Maharshi, Raajarshi, and Devarshi --
--Maharshi are those ones who are born from Pitaamaha Brahmaa Jee, such as Atri, Vashishth, Pulah etc;
--Raajarshi are those kings on Prithvi who have attained Rishi like qualities like Janak etc kings; and
--Devarshi are those Devtaa who have attained Rishi like qualities, such as Devarshi Naarad Jee.

Brahmarshi - But we have heard the name of "Brahmarshi" also. Vishwaamitra Jee wanted only to be Brahmarshi, because Vashishth Jee was also a Brahmarshi and he wanted to be equal to Vashishth Jee. Brahmarshi are those who have known Brahm.

Raajarshi
Raajarshi are those kings who have Rishi like qualities. The most famous Raajarshi is Raajaa Janak (Seetaa's father).

Devarshi
Devarshi is that Devtaa who has Rishi like qualities. Since they come under the category of Devtaa, they can travel in all the three Lok - Swarg Lok, Prithvi Lok and Paataal Lok. The most famous and known Devarshi is Naarad Jee

Maharshi
The Rishi who have been born from Brahmaa Jee are called Maharshi, so his all the 9 sons  - Atri, Pulah, Pulastya, Mareechi etc are Maharshi

Brahmarshi
Brahmarshi is that Rishi who has known Brahm and has reached at the highest intelligence level, such as Vashishth Jee