Most of the textual differences are ironed out in the source language which are not especially accessible. English bibles are not translated directly from manuscripts but from one or more "critical texts" which are scholarly reconstructions of the underlying manuscripts. A common one is Nestle-Aland [0], you can read it online. The smaller text appearing there are variant readings that have some degree of manuscript support.

The vast majority of them are terribly boring, by way of example, the very first one on the page:

> τοῦτο δὲ ὅλον γέγονεν ἵνα πληρωθῇ τὸ ῥηθὲν ὑπὸ κυρίου διὰ τοῦ προφήτου λέγοντος ἰδοὺ ἡ παρθένος ἐν γαστρὶ ἕξει καὶ τέξεται υἱόν, [καλέσουσιν τὸ ὄνομα αὐτοῦ Ἐμμανουήλ] ὅ ἐστιν μεθερμηνευόμενον [μεθ’ ἡμῶν ὁ θεός.]

> All this took place to fulfill what the Lord had spoken by the prophet: Behold, the virgin in the womb will conceive, and will bear a son, and [they will call the name of him Immanuel], which is translated, '[with us God]'

The [bracketed] parts are omitted in some manuscripts. These omissions may feel significant, but since this is a quotation of well-known saying, the shorter readings are probably some scribe trying to scrimp on paper who thinks you should go buy that book if you wanted to know what's in it. This can be annoying in cases where the book in question didn't make it out to modern scholarship, and all the scribes scrimped on paper that day, which does happen.

Less commonly (although not infrequently), a textual problem is significant enough to bleed all the way through to the end user. This typically happens when there is no agreement about how to resolve the conflicts or the possibilities are so strange as to be nonsensical. One amusing example is 1 sam 6:19 [1]. There is overwhelming manuscript support for a reading like this:

> But the Lord struck down some of the people of Beth Shemesh because they had looked into the ark of the Lord; he struck down 50,070 of the men. [NET]

However, in addition to there being a glaring grammar error in the manuscript, this reading presents a number of obvious problems: logistical (how exactly do you get 50,070 people to all look into a box?), numerical (this is a surprisingly exact figure to collect during wartime in enemy territory), internal (elsewhere in the same book Beth Shemesh is a tiny rural village), archeological (it really was a tiny rural village).

This problem has lead to various wildly different resolutions over the years, a small sampler:

> But God struck down some of the inhabitants of Beth Shemesh, putting seventy of them to death because they looked into the ark of the LORD. The people mourned because of the heavy blow the LORD had dealt them. [NIV]

> God struck down the men of Beth-shemesh because they looked inside the ark of the LORD. He struck down 70 men out of 50,000 men. The people mourned because the LORD struck them with a great slaughter. [HCSB]

> Then God smote those of Bethshemesh because they had looked at the ark of the LORD; he smote fifty thousand of the people and seventy principal men. And the people lamented because the LORD had smitten the people with such a great slaughter. [Jubilee]

> And He smiteth among the men of Beth-Shemesh, for they looked into the ark of Jehovah, yea, He smiteth among the people seventy men -- fifty chief men; and the people mourn, because Jehovah smote among the people -- a great smiting. [YLT]

> Perhaps the text should be understood to read the LORD killed 70 men and 50 oxen. [NKJ footnote]

[0] http://www.nestle-aland.com/en/read-na28-online/

[1] http://biblehub.com/1_samuel/6-19.htm