Moses addresses Deuteronomy 12 to Israel on the east side of the Jordan River. He tells them that, instead of having altars and high places throughout the country, they should bring their required and free-will offerings, other sacrifices, firstlings, and tithes (plural in verses 6, 11) to the place where God put His name. There, the people were to eat before Him.
The second tithe here is in addition to the tithe commanded in Numbers 18. It was to be eaten by the individual Israelite, but not at home ("not . . . within your gates," Deuteronomy 12:17). This tithe was not to come out of the first tithe, because the latter was the Levites' inheritance. The Levites lived off the first tithe in all their cities throughout the year. Therefore, the first tithe that they normally lived on cannot have been the second tithe that God forbids all to eat in their homes. The Bible nowhere states that the Levite had to bring a special tithe to eat at the feasts (such as the Feast of Tabernacles). The people were to bring it. Verse 18 specifically states that the Levite should also eat of the second tithe where the Lord chose to place His name (see Numbers 18:21-24).
In Deuteronomy 12:11, the plural usage shows that Moses distinctly refers to more than one tithe. Deuteronomy 14:23 uses "tithe" in the singular because the passage describes only the specific tithe that we are to consume where God places His name.