The word apistos is rendered in a number of different ways, but all of them relate to the sin of unbelief.
Paul refers to those "that believe not" in the context of marriage. A Christian spouse is not bound if he or she has a husband or wife who doesn't believe, and who does not want to remain married (1 Cor. 7:12,13). The same word is rendered as unbelieving three times in the next two verses (1 Cor. 7:14-15), still referring to the non-Christian spouse.
If someone who "believes not" invites a Christian to a feast, he should certainly feel free to go (1 Cor. 10:27).
The gift of tongues was given for those who "believe not" (1 Cor. 14:22, 24), while prophesy is for believers. We can see the impact of a right use of these gifts if an unbeliever comes into the service (1 Cor. 14:23).
The word is translated as unbelieving in two places. Nothing is pure for the defiled and unbelieving (Tit. 1:15). The lake of fire is reserved for, among others, the unbelieving (Rev. 21:8).
Another rendition is faithless, and we have one example of this from each gospel. When His disciples could not heal a demon-possessed boy, Jesus lamented His perverse and faithless generation (Matt. 17:17; Mk. 9:19; Luke 9:41). In the gospel of John, Jesus told Thomas not to be faithless, but rather to believe (John 20:27).
The word can also mean unbeliever. The servant who abuses the others will be assigned a portion with the unbelievers (Luke 12:46). The apostle was appalled that the Corinthian Christians were willing to go to law against each other before unbelievers (1 Cor. 6:6). We are not to be yoked together with unbelievers (2 Cor. 6:14). The same word is rendered as infidel in the next verse (2 Cor. 6:15). And a man who does not care for his own family is worse than an infidel (1 Tim. 5:8).
The god of that age had blinded the minds of those who had not believed (2 Cor. 4:4).
And in one place Paul uses the word to refer to a "thing incredible" (Acts 26:8), referring to unbelief in the resurrection.