What does the beast represent?
From the New Testament seminary manual:
[The] beast described in verse 3 may represent Rome in John’s day as well as corrupt kingdoms and nations in the last days (see Revelation 17:8–13).
Further explanation can be found from the New Testament Student Manual
Scholars have historically interpreted the beast with seven heads as an allusion to the city of Rome because Rome was founded upon seven hills, as well as a reference to a number of Roman emperors in the first century A.D. Though these are possible meanings, a broader interpretation may be that Rome represented a seat of corruption in John’s day. As such, this worldly city may be likened to similar centers of corruption in all eras.
Earlier in Revelations 12:3,9 there is a red dragon with seven heads and ten horns which represents the Devil/Satan, this could be the same beast.
Whom does the woman symbolize?
The seminary manual references verse 18:
18 And the woman which thou sawest is that great city, which reigneth over the kings of the earth.
And then explains:
[The] "great city" (verse 18) refers to spiritual Babylon (see D&C 133:14). Because of the worldliness and corruption of ancient Babylon, and because it was a place where the children of Israel were captive, Babylon is often used figuratively in the scriptures to represent sin, worldliness, the influence of the devil on the earth, and spiritual captivity (see also 1 Nephi 13:1–9; 14:9–10).
From the student manual:
she is the "mother of harlots," which indicates that she gives birth to other prostitutions—organizations, governments, and ideologies that spew forth wickedness
On the verses as a whole the student manual states:
One possible meaning of John’s description in these verses is that in the last days, a lifestyle of sexual immorality, wealth, and violence would permeate the world. Institutions, governments, and people who embrace this lifestyle can be seen as part of Babylon.
On Babylon Elder David R. Stone says:
There is no particular city today which personifies Babylon. Babylon was, in the time of ancient Israel, a city which had become sensual, decadent, and corrupt. ...
That sensuality, corruption, and decadence, and the worshipping of false gods are to be seen in many cities, great and small, scattered across the globe.
