Let's start with the ayat/verses outlining Allah (swt)'s commands on us:
"And rule between them by that which Allah revealed to you, and do not follow their vain desires away from the truth which came to you." (Qur'an 5:48)
And rule between them by that which Allah revealed to you and do not follow their whims, and beware (be on the alert) that they may deviate you away from even some part of what Allah revealed to you." (Qur'an 5:49)
Then onto a few ahadith of Rasul'Allah (saw) on the subject:
"The Prophets ruled over the children of Israel, whenever a Prophet died another Prophet succeeded him, but there will be no Prophet after me. There will be Khulafaa' and they will number many.' They asked: ‘What then do you order us?' He said: ‘Fulfil the Bay'ah to them one after the other and give them their due. Surely Allah will ask them about what He entrusted them with.'" (Muslim)
"Whosoever takes off his hand from allegiance to Allah (swt) will meet Him (swt) on the Day of Resurrection without having any proof for him, and whoso dies whilst there was no Bay'ah (allegiance) on his neck (to a Khaleefah), he dies a death of Jahiliyyah (ignorance).'" (Muslim)
Then the views of the classical fuqahah (scholars):
“They (scholars of ahl al-sunnah) said regarding the khilafah and imamah that the Imamah is an obligation obligated on the ummah to the end of establishing the imam who would appoint for them judges and ministers, secure their frontiers, mobilise their armies, divide the fay’, and give justice to the oppressed from the oppressors; and they said that the way of contracting the imamah for the ummah is by their choice through exertion of effort.” - Abd al-Qahir al Baghdadi (died in 1037 CE) in al-Farq bayn al-Firaq, page 340.
“All of ahl ul-sunnah agreed, as did all the murji’a, all the shi’a, and all the khawarij upon the obligation of the Imamah, and that it is obligatory on the ummah to submit to a just imam, who establishes upon them the ahkam of Allah, and manages their affairs by the ahkam of the shari’a with which the Messenger of Allah (saw) came; except only the najadat from the khawarij who said: the people are not obliged with the imamah, rather what is upon them is to mutually practice the what is correct between them.” ibn Hazm (died 1064) in al-Fasl fi Milal wa 'l-Ahwaa' wa 'l-Nihal' 4:87
“The Imams (of the four madhabs: Abu Hanifa, Malik, Shaf’i, Ahmad) - may Allah have mercy on them – all consented that the Imamah is an obligation, and that the Muslims must appoint an imam who would implement the rites of the deen, and give the oppressed justice against the oppressors, and they agreed that it is not permitted that there be over the Muslims, at any one time, two imams, in agreement or discord, and that the imams are from Quraysh and that is permitted for the imam to choose a successor.” Imam al-Juzayri (died in 1360) in al-Fiqh ‘ala al-Mathahib al-Arba’a, 5:416
Secondly, as for the definition of "ummah."
The linguistic definition of "ummah" is 'nation' or 'community.' The term is referring to a set of people, irrespective of usage. In the Qur'an and Sunnah, we've seen specific mentions made to the 'ummah' of Muhammad (saw). In broad terms this could include everyone who came during and after the time of the Prophet (saw) (Muslims and non-Muslims), but in other contexts, it is referring to the people who accepted Muhammad (saw) as the seal of prophets and messengers, i.e. Muslims.
In the Qur'an Allah (swt) says to the Muslims:
"You are the best ummah produced [as an example] for mankind. You enjoin what is right and forbid what is wrong and believe in Allah." (Qur'an: 3:110)