The beast appears in Daniel as well. It was a common allegorical way to represent the pagan empires that oppressed the true believers. The beast referenced in Rev. 13:1, with the usual 7 heads and 10 horns, is the Roman Empire. "The name of blasphemy" on its heads refers to worshiping the emperors as gods, which was really just a formality in public rituals intended to bind together the multicultural diversity of the empire, in much the same way as some nations salute their flags and recite pledges of allegiance. Christian refusal to take part in it made them objects of suspicion.
The Beast from Vs:18 is not the very same Beast that is in Vs:1, but they (18) are the two that end up in the lake at Christ's return. The Beast of Vs:1 is a fallen angel, that is what is released from the pit when the 5th trump sounds. Satan is the star that falls from Heaven, the beginning of what are the first two woes.
Rome is the 6th king, this Beast is the 8th. Satan is the one that opens the pit, that makes him the 7th. Rome is the one 'that is', the one that had rule over Jerusalem.
Re:17:10:
And there are seven kings:
five are fallen,
and one is,
and the other is not yet come;
and when he cometh,
he must continue a short space.
Re:17:11:
And the beast that was,
and is not,
even he is the eighth,
and is of the seven,
and goeth into perdition.
(before the flood he was, is not now because he is in the pit, will be again when Satan is given a key that releases him)
13:1- 7 heads & blasphemy, 7 characteristics that Satan has taught man since the fall, each and everyone is an abomination to God's way of how things should be done. Re:21:8 is a list of what God would 'fight against'.
-The 10 horns are this Beasts human 'helpers', horn could specify language (10 different languages) or it is just a reference to that 10 being able to make noise.
Re:17:12:
And the ten horns which thou sawest are ten kings,
which have received no kingdom as yet;
but receive power as kings one hour with the beast.
13:2-a reference to the beasts in, the 3 covered were armies of men, it does not cover the beast of Da:7:7, that is the iron of the statue, the clay of that statue is the 10 men that align themselves with this beast. They are not kingdoms before that 1 hour (so none can yet be named) but they have kingdoms after that hour. Satan has power over this Beast, same as he was the most powerful of all the fallen angels that existed with man before the flood.
13:3-the head with the deadly wound would be whatever sin the fallen angel committed that got him into the pit. That same (blasphemy) can again be practiced for the time he is 'doing things', wound healed.
13:4-Both Satan and him will be worshiped, some willingly, some through fear of death. Nobody will see them as being 'defeatable'.
To save space the next few verses are also about this same fallen angel, including the death of the two witnesses and the amount of time he has in total. So when it says worshiped above that applies right up to his demise.
Vs:11 is when the False Prophet is with this same Beast from the previous verses.
This is when Daniel put him as coming into power,
Da:7:8:
I considered the horns,
and,
behold,
there came up among them another little horn,
before whom there were three of the first horns plucked up by the roots:
and,
behold,
in this horn were eyes like the eyes of man,
and a mouth speaking great things.
Again this is shortened to save space.
Until relatively recently it was common to use letters of the alphabet to signify numbers, and there are still some people who do it. Jews, Greeks, and Romans all did it; Roman numerals are only the most obvious case. In Greek and Hebrew, every letter was assigned a numerical value, so of course any name made up of those letters could also be assigned a numerical value. Mystics assumed the words of the Bible must have numerical significance as well, and they spent a lot of time and energy trying to tease out meanings from them. It's called "gematria," a minor corruption of the Greek word that gave us "geometry." The number of the beast is the only significant example of gematria in the Bible.
One quick question, everybody, including me, sees his number as being 666. As obscure as God can be at times, can "Six hundred threescore and six" be written as 600606?
I hope this is clear and short enough.