Madinah is the city in which the Prophet Muhammad (peace be upon him) migrated to from Mecca.
Madinah was the city in which the Muslim community grew and many of the verses of the Quran were revealed which dealt with religious rulings dealing with everyday life including business transactions, the prohibition of alcohol, the head covering for women, and other social and ethical rulings.
Madinah is the city in which the Prophet Muhammad (peace be upon him) is buried and a city which he loved
Narrated Anas: Whenever the Prophet returned from a journey and observed the walls of Medina, he would make his Mount go fast, and if he was on an animal (i.e. a horse), he would make it gallop because of his love for Medina. (Bukhari)
Narrated Anas: (The people of) Bani Salama intended to shift near the mosque (of the Prophet) but Allah's Apostle disliked to see Medina vacated and said, "O the people of Bani Salama! Don't you think that you will be rewarded for your footsteps which you take towards the mosque?" So, they stayed at their old places. (Bukhari)
Narrated Anas: The Prophet said, "O Allah! Bestow on Medina twice the blessings You bestowed on Mecca." (Bukhari)
It is highly recommended for Muslims to visit Madinah
The Prophet (peace and blessings of Allah be upon him) who said: “No journey should be undertaken to any mosque except three: al-Masjid al-Haraam, this mosque of mine and al-Masjid al-Aqsa.” (Sahih al-Bukhari, Sahih Muslim)
Madinah has special status and rules that Muslims should follow:
Narrated Anas: The Prophet said, "Medina is a sanctuary from that place to that. Its trees should not be cut and no heresy should be innovated nor any sin should be committed in it, and whoever innovates in it a heresy or commits sins (bad deeds), then he will incur the curse of Allah, the angels, and all the people." (See Hadith No. 409, Vol 9). (Bukhari)
Narrated Abu Huraira: The Prophet said, "I have made Medina a sanctuary between its two (Harrat) mountains." The Prophet went to the tribe of Bani Haritha and said (to them), "I see that you have gone out of the sanctuary," but looking around, he added, "No, you are inside the sanctuary." (Bukhari)
Narrated 'Ali: We have nothing except the Book of Allah and this written paper from the Prophet (where-in is written :) Medina is a sanctuary from the 'Air Mountain to such and such a place, and whoever innovates in it an heresy or commits a sin, or gives shelter to such an innovator in it will incur the curse of Allah, the angels, and all the people, none of his compulsory or optional good deeds of worship will be accepted. And the asylum (of protection) granted by any Muslim is to be secured (respected) by all the other Muslims; and whoever betrays a Muslim in this respect incurs the curse of Allah, the angels, and all the people, and none of his compulsory or optional good deeds of worship will be accepted, and whoever (freed slave) befriends (take as masters) other than his manumitters without their permission incurs the curse of Allah, the angels, and all the people, and none of his compulsory or optional good deeds of worship will be accepted. (Bukhari)