If you don’t believe in the God of Abraham you are, for all intents and purposes an idolater.
The Verse of the Sword states that only an immediate conversion followed by a recitation of the mandatory daily prayers and a charitable donation will spare the life of an idolater or save him, or her, from a life in captivity as the slave of a believer.
Having said that, there are three overriding conditions where an unrepentant idolater’s life is not yours for the taking at your discretion:
1) during the sacred months if they don’t pose a threat, real or imagined
2) if you have a compact i.e. treaty with a group of them, or
3) if they seek refuge with you for the purpose of learning about this Allah.
9 At- Tawbah
(The surah Repentance is mainly about war, a reason for the lack of the ubiquitous formula-invocation “in the name of Allah, the Compassionate, the Merciful”.)
9:1 This is an immunity from Allah and His Messenger to those idolaters with whom you made compacts.
9:2 Travel, then, in the land freely for four months, and know that you will never able to thwart Allah, and that Allah shall disgrace the unbelievers.
9:3 This is a proclamation from Allah and His Messenger to mankind on the day of the great pilgrimage (10th of Dhul Hijjah), that Allah is absolved of the idolaters, as is His Messenger. If you repent, it will be better for you; but if you turn away, know that you shall never thwart Allah. Proclaim to those who disbelieve a grievous punishment.
9:4 Except for those idolaters with whom you made a compact, then they did not fail you in anything and did not lend support to anybody against you. Honour your compact with them until the end of its term. Allah loves the righteous.
9:5 Then, when the Sacred Months (four months during which war was prohibited in pre-Islamic time) are over, kill the idolaters wherever you find them, take them [as captives], besiege them, and lie in wait for them at every point of observation. If they repent afterwards, perform the prayer and pay the alms, then release them. Allah is truly All-Forgiving, Merciful.
An idolater can seek refuge with the believers for the purpose of learning about Allah, and it must be granted. If after learning about Allah the idolater is still not convinced that submitting to His Will is something he wants to do, the believers are expected to return him unharmed to where they cannot immediately harm him i.e. “his place of security”.
9:6 And if any one of the idolaters should seek refuge with you, give him refuge, so that he may hear the Word of Allah; then convey him to his place of security. That is because they are a people who do not know.
This revelation caused much consternation among the believers as the following revelations will attest.
9:7 How can the idolaters have a compact with Allah and His Messenger, except for those you made a compact (a formal treaty) with at the Sacred Mosque? So long as these honour their obligations to you, honour yours to them. Allah loves the righteous.
Everyone knows that an idolater’s word means nothing; that they are sinners and given the opportunity will probably kill you.
9:8 How [can that be]? If they overcome you, they will observe neither kingship nor compact with you. They only give you satisfaction with their mouths, while their hearts refuse, and most of them are sinners.
9:9 They have sold Allah’s Revelations for a small price, and have barred [others] from His Path. Evil indeed is what they do!
9:10 They observe with the believers neither kingship nor compact. Those are the real transgressors.
9:11 Yet, if they repent, perform the prayer and pay the alms, they will be your brethren in religion. We expound the revelations to a people who know.
If they break their oath, you know what to do!
9:12 But if they break their oaths after their pledge [is made] and abuse your religion, then fight the leaders of unbelief; for they have no regards for oaths, and that perchance they may desist.
9:13 Will you not fight a people who broke their oaths and intended to drive the Messenger out, seeing that they attack you first? Do you fear them? Surely, you ought to fear Allah more, if you are real believers.
9:14 Fight them, Allah will punish them at your hands, will disgrace them, give you victory over them, and heal the hearts of a believing people.
9:15 And He will remove the rage from their hearts. Allah shows mercy to whomever He pleases, and Allah is All-Knowing, Wise.
Christians and Jews
For the Jews and Christians and the illusive Sabians it’s a little more complicated. If you are a Jew or a Christian and, as unlikely as it would seem, you don’t believe in the God of Abraham, then you are no better than an idolater and not converting is not an option.
Unbelievers who believe, and only believe in the God of Abraham (the so-called People of the Book) are not to be killed “wherever you find them” if they are submissive, and are willing to humbly pay money in the form of a poll-tax (a tax levied on people rather than on property) so that their lives might be spared.
9:29 Fight those among the People of the Book who do not believe in Allah and the Last Day, do not forbid what Allah and His Messenger have forbidden and do not profess the true religion, till they pay the poll-tax out of hand and submissively.
Allah quickly reminds the believers, in the verse immediately following 9:29, that while He is willing to spare the lives of Christians and Jews ready-and-able to dutifully pay his life-saving-tax while acknowledging His pre-eminence, they are still a perverted people for saying that Allah has a son after he told them not to.
9:30 The Jews say: “Ezra is the son of Allah”, and the Christians say: “The Messiah is the son of Allah.” That is their statement, by their mouths; they emulate the statement of the unbelievers of yore. May Allah damn them; how they are perverted!
9:31 They take their rabbis and monks as lords besides Allah, as well as the Messiah, son of Mary, although they are commanded to worship none but One God. There is no god but He; exalted He is above what they associate with Him.
And why would they do this?
9:32 They wish to put out Allah’s Light with their mouths, and Allah allows nothing less than perfecting His Light, even if the unbelievers should resent it.
9:33 It is He Who sent His Messenger with the guidance and the true religion, in order to make it triumph over every religion, even if the polytheist should resent it.
And then, there are times when a bloody beating, some time spent in chains and a ransom is all that is needed. Killing is optional.
47:4 So when you meet the unbelievers, strike their necks till you have bloodied them, then fasten the shackles. Thereupon, release them freely or for a ransom, till the war is over. So be it. Yet had Allah wished, He would have taken vengeance upon them, but he wanted to test you by one another. Those who die in the Cause of Allah, He will not render their works perverse.
The more verbose Mohsin Khan translation is unambiguous as to what “strike their necks” is all about, among other things.
47:4 So, when you meet (in fight - Jihad in Allah's Cause) those who disbelieve, smite (their) necks till when you have killed and wounded many of them, then bind a bond firmly (on them, i.e. take them as captives). Thereafter (is the time) either for generosity (i.e. free them without ransom), or ransom (according to what benefits Islam), until the war lays down its burden. Thus [you are ordered by Allah to continue in carrying out Jihad against the disbelievers till they embrace Islam and are saved from the punishment in the Hell-fire or at least come under your protection], but if it had been Allah's Will, He Himself could certainly have punished them (without you). But (He lets you fight) in order to test some of you with others. But those who are killed in the Way of Allah, He will never let their deeds be lost.
47:5 He shall guide them and set their minds aright;
47:6 And shall admit them into Paradise which He has made known to them.
47:7 O believers, if you support Allah, He will support you and steady your footsteps.
47:8 But as to the unbelievers, wretched are they and perverse are their works.
47:9 That is because they despised what Allah has sent down; so He foiled their actions.
I have killed the likes of them before and I will do so again!
47:10 Did they not travel in the land and see what was the fate of those who preceded them? Allah brought utter destruction on them; and the like of this awaits the unbelievers.
47:11 That is because Allah is the Protector of the believers, but the unbelievers shall have no protector.