The Messenger (peace and blessings of Allah be upon him) is the best and most perfect of creation, and the most beloved to Allah and the most honoured by Him. But this does not mean that his human characteristics are to be denied, or that any of the duties of worship which are due only to Allah should be directed to him. The Messenger (peace and blessings of Allah be upon him) was a human being who suffered from sickness and real death the same as any other human being. Allah said (interpretation of the meaning): 
“Verily, you (O Muhammad) will die, and verily, they (too) will die” [al-Zumar 39:30] 

“And We granted not to any human being immortality before you (O Muhammad); then if you die, would they live forever?” [al-Anbiyaa’ 21:34]
The Messenger (peace and blessings of Allah be upon him) died and was buried in his grave, hence al-Siddeeq Abu Bakr (may Allah be pleased with him) said: “Whoever used to worship Muhammad, Muhammad has died, but whoever used to worship Allah, Allah is alive and will never die.”

Undoubtedly, prophets are alive in their graves in a way that is known only to Almighty Allah. The Prophet Muhammad (peace and blessings be upon him) is reported to have said, "Allah has forbidden the earth to consume the bodies of the Prophets" (Abu Dawud).

Responding to the question, Sheikh Muhammad Saleh Al-Munajjid, a prominent Saudi Islamic lecturer and writer, states the following:

Indeed, there are many hadiths that state that the earth doesn't consume the bodies of prophets; they are alive in their graves. The Prophet Muhammad (peace and blessings be upon him) is reported to have said, "The best of your days is Friday, so send more blessings upon me on that day, for your blessings will be shown to me." They said, "O Messenger of Allah, how will they be shown to you when you (your body) have disintegrated?" He said, "Allah has forbidden the earth to consume the bodies of the prophets" (Abu Dawud).

Thus, the prophets are dead as far as the people of this world are concerned. Referring to this, Allah Almighty says [Lo! thou wilt die, and lo! they will die] (Az-Zumar 39:30).

However, before Allah the prophets are alive, for if the martyrs are alive with Allah, then the prophets are undoubtedly higher in status before Allah (Fath Al-Bari, 6/444).

Moreover, the Prophet Muhammad (peace and blessings be upon him) is reported to have said, "The prophets are alive in their graves and they pray" (Al-Bazzar). Muslim scholars state that the prayer referred to in the hadith is something that they are blessed with just as the people of Paradise are blessed with tasbih (glorifying Allah).

Based on what is mentioned above, we can conclude that the prophets are alive in their graves in a way known only to Almighty Allah.

They (peace and blessings be upon them all) are alive in their graves, apart from `Isa (Jesus, peace be upon him), whom Allah raised up into heaven, as He says in the Qur'an: [And because of their saying: We slew the Messiah Jesus son of Mary, Allah's messenger. They slew him not nor crucified, but it appeared so unto them; and lo! those who disagree concerning it are in doubt thereof; they have no knowledge thereof save pursuit of a conjecture;  they slew him not for certain] (An-Nisaa ' 4:157).

And Allah Knows Best.