In Exodus chapter 20, and Deuteronomy chapter 5, God gives the Ten Commandments. In both cases, they are the same, there are clearly ten of them, and they are even given in the same order:

1. Thou shalt have no other gods before me (Exodus 20:3, Deuteronomy 5:7)
2. Thou shalt not make thee any graven image (Exodus 20:4-6, Deuteronomy 5:8-10)
3. Thou shalt not take the name of the Lord thy God in vain (Exodus 20:7, Deuteronomy 5:11)
4. Remember the sabbath day, to keep it holy (Exodus 20:8-11, Deuteronomy 5:12-15)
5. Honour thy father and thy mother (Exodus 20:12, Deuteronomy 5:16)
6. Thou shalt not kill (Exodus 20:13, Deuteronomy 5:17)
7. Thou shalt not commit adultery (Exodus 20:14, Deuteronomy 5:18)
8. Thou shalt not steal (Exodus 20:15, Deuteronomy 5:19)
9. Thou shalt not bear false witness against thy neighbour (Exodus 20:16, Deuteronomy 5:20)
10. Thou shalt not covet anything that is thy neighbour's (Exodus 20:17, Deuteronomy 5:21)

However, after Moses breaks the original stone tablets that God made in Exodus 32:19, he makes another set of tablets, and gets the commandments again from God (Exodus 34:4-28).

The alleged contradiction is that the content of this section does not match the Ten Commandments, which is correct:

Exodus 34:11-28

11 Observe thou that which I command thee this day: behold, I drive out before thee the Amorite, and the Canaanite, and the Hittite, and the Perizzite, and the Hivite, and the Jebusite.
12 Take heed to thyself, lest thou make a covenant with the inhabitants of the land whither thou goest, lest it be for a snare in the midst of thee:
13 But ye shall destroy their altars, break their images, and cut down their groves:
14 For thou shalt worship no other god: for the Lord, whose name is Jealous, is a jealous God:
15 Lest thou make a covenant with the inhabitants of the land, and they go a whoring after their gods, and do sacrifice unto their gods, and one call thee, and thou eat of his sacrifice;
16 And thou take of their daughters unto thy sons, and their daughters go a whoring after their gods, and make thy sons go a whoring after their gods.
17 Thou shalt make thee no molten gods.
18 The feast of unleavened bread shalt thou keep. Seven days thou shalt eat unleavened bread, as I commanded thee, in the time of the month Abib: for in the month Abib thou camest out from Egypt.
19 All that openeth the matrix is mine; and every firstling among thy cattle, whether ox or sheep, that is male.
20 But the firstling of an ass thou shalt redeem with a lamb: and if thou redeem him not, then shalt thou break his neck. All the firstborn of thy sons thou shalt redeem. And none shall appear before me empty.
21 Six days thou shalt work, but on the seventh day thou shalt rest: in earing time and in harvest thou shalt rest.
22 And thou shalt observe the feast of weeks, of the firstfruits of wheat harvest, and the feast of ingathering at the year's end.
23 Thrice in the year shall all your menchildren appear before the Lord God, the God of Israel.
24 For I will cast out the nations before thee, and enlarge thy borders: neither shall any man desire thy land, when thou shalt go up to appear before the Lord thy God thrice in the year.
25 Thou shalt not offer the blood of my sacrifice with leaven; neither shall the sacrifice of the feast of the passover be left unto the morning.
26 The first of the firstfruits of thy land thou shalt bring unto the house of the Lord thy God. Thou shalt not seethe a kid in his mother's milk.
27 And the Lord said unto Moses, Write thou these words: for after the tenor of these words I have made a covenant with thee and with Israel.
28 And he was there with the Lord forty days and forty nights; he did neither eat bread, nor drink water. And he wrote upon the tables the words of the covenant, the ten commandments.

What people construing this as a contradiction ignore is basically the entirety of the events that took place between Exodus chapter 20 and Exodus chapter 34.

The whole of Exodus chapters 21-23 consists of more commandments. These were also written down (Exodus 24:4), perhaps alongside a copy of the Ten Commandments that Moses himself made, or are in addition to the tablets that God himself would give Moses later (Exodus 31:18). Moses then confirms the covenant with the people (Exodus 24:7-8), and then goes back up the mountain for another 40 days (Exodus 24:18), where he receives further instructions about the Law, which occupies all of Exodus chapters 25-31.

So, what we are seeing in Exodus 34:11-28 is a mix of some of the Ten Commandments, alongside some supplementary commandments like the ones Moses received in Exodus 21-23 - basically an extremely brief summary so as to not repeat what God had just told Moses for the better part of the last 14 chapters, for the purpose of communicating that God was giving the commandments to Moses again.