Who are the twenty-four elders?
The twenty four elders around God's throne are glorified men, the spirits of just men made perfect who lived before Christ, not angelic beings and not the raptured Church.
Scripture describes the twenty four elders as seated on thrones around God, clothed in white raiment, with crowns of gold on their heads. They are called elders, which tells us plainly that they are men, not angelic beings. Since members of the New Testament Church do not receive their crowns and white raiment until the rapture, these elders cannot be part of the Church. Instead they are identified as the spirits of just men made perfect, faithful people such as Abraham, Job, and the prophets who lived and died before Christ came.
When Christ rose from the dead, He descended into the lower parts of the earth and brought these Old Testament saints into perfection, and many of their bodies were raised at that time. Hebrews later speaks of this same company as a great cloud of witnesses, now dwelling in God's presence as living proof that death cannot overturn the promises of God. The twenty four elders, then, are honored representatives chosen from this company of the redeemed dead of former ages, seated in glory before God's throne.
Their individual names are never revealed in Scripture, so it is not right to guess who each one specifically is. What is confirmed is their identity and role: in Revelation 5, they fall down with harps and golden bowls of incense, sing a new song about being redeemed by the blood of the Lamb, and are shown to reign as kings and priests unto God, further confirming that they are glorified redeemed men rather than angels.
The scriptures
And round about the throne were four and twenty seats: and upon the seats I saw four and twenty elders sitting, clothed in white raiment; and they had on their heads crowns of gold.
But ye are come unto mount Sion, and unto the city of the living God, the heavenly Jerusalem, and to an innumerable company of angels, To the general assembly and church of the firstborn, which are written in heaven, and to God the Judge of all, and to the spirits of just men made perfect, And to Jesus the mediator of the new covenant, and to the blood of sprinkling, that speaketh better things than that of Abel.
Henceforth there is laid up for me a crown of righteousness, which the Lord, the righteous judge, shall give me at that day: and not to me only, but unto all them also that love his appearing.
He that overcometh, the same shall be clothed in white raiment; and I will not blot out his name out of the book of life, but I will confess his name before my Father, and before his angels.
(Now that he ascended, what is it but that he also descended first into the lower parts of the earth? He that descended is the same also that ascended up far above all heavens, that he might fill all things.)
And the graves were opened; and many bodies of the saints which slept arose, And came out of the graves after his resurrection, and went into the holy city, and appeared unto many.
And I beheld, and I heard the voice of many angels round about the throne and the beasts and the elders: and the number of them was ten thousand times ten thousand, and thousands of thousands; Saying with a loud voice, Worthy is the Lamb that was slain to receive power, and riches, and wisdom, and strength, and honour, and glory, and blessing. And every creature which is in heaven, and on the earth, and under the earth, and such as are in the sea, and all that are in them, heard I saying, Blessing, and honour, and glory, and power, be unto him that sitteth upon the throne, and unto the Lamb for ever and ever. And the four beasts said, Amen. And the four and twenty elders fell down and worshipped him that liveth for ever and ever.
From the books
This answer is drawn from these passages in the Mary Banks Faith Library.