I read a Hadīth about the Ummah of the Prophet (sws) being divided into 73 sects, and that only one will go to heaven. I wanted to know the definition of a sect, and the one which is on the right, and why?

By collecting and analyzing all the texts of the Hadīth you have referred to, the picture which emerges is that:

i) The prediction stated in these Ahādīth pertain to the period just after the Prophet (sws), and relate to events within the land of Arabia. Such Ahādīth  are meant to put Muslims on guard, and are also a means to substantiate the Prophethood of Muhammad (sws) for people of his times.

ii) The number 73 is not been used here in its literal sense. It is used here figuratively to emphasize a point. Such a usage has examples in every language. For example we say: If Ahmad has faltered three times you will falter four times. This does not mean that we are counting the number of times some one is going to falter. It only means that we are warning our addressee that he will falter more than Ahmad and therefore he should be very careful.

iii) The word "firqah" here does not imply a religious sect. It means the various political factions and groups which do not submit to the authority of the state and create anarchy in the society.

iv) These Ahādīth also direct Muslims to remain attached to the "al-jama'ah", when such a time of anarchy comes. The word "al-jama`ah" means the state and does not mean a religious faction or group.

v) It is known that many factions like the Khawārij and the Sabāi created a lot of nuisance in the period after the Prophet (sws) by rebelling against the state authority. It is groups such as these which have been referred to in such Ahādīth. The Prophet (sws) is urging his companions and followers not to become part of such rebellious factions and to cling to the authority of the state.