What does the Bible tell us about how people knew right from wrong before the Law of Moses?

In the Garden of Eden we find that there was: http://en.wikipedia.org/wiki/Tree_of_the_knowledge_of_good_and_evil

Humans ate from it. About this tree God said:

יז וּמֵעֵץ, הַדַּעַת טוֹב וָרָע--לֹא תֹאכַל, מִמֶּנּוּ: כִּי, בְּיוֹם אֲכָלְךָ מִמֶּנּוּ--מוֹת תָּמוּת. 17 but of the tree of the knowledge of good and evil, thou shalt not eat of it; for in the day that thou eatest thereof thou shalt surely die.'

It is a curious situation, because the humans are punished for a sin that they committed before they ate from the tree of Knowledge of Good and Evil. There are several words for "knowing" in Hebrew but the most clear one is "daat", and in this case that is what's used in הַדַּעַת (knowing).

So humans are told the consequences of their action, but it may be said that they did not know the actual reasons for WHY their action is bad. Merely that it was prohibited.

After they ate from the tree, it was said of them:

וַתִּפָּקַחְנָה, עֵינֵי שְׁנֵיהֶם, וַיֵּדְעוּ, כִּי עֵירֻמִּם הֵם; וַיִּתְפְּרוּ עֲלֵה תְאֵנָה, וַיַּעֲשׂוּ לָהֶם חֲגֹרֹת. 7 And the eyes of them both were opened, and they knew that they were naked; and they sewed fig-leaves together, and made themselves girdles.

Now they are able to understand on their own what is desirable and what is not. This is illustrated by their distaste for being naked, without anyone telling them this. They also seem to become ashamed of their state and their actions, and blame each other:

י וַיֹּאמֶר, אֶת-קֹלְךָ שָׁמַעְתִּי בַּגָּן; וָאִירָא כִּי-עֵירֹם אָנֹכִי, וָאֵחָבֵא. 10 And he said: 'I heard Thy voice in the garden, and I was afraid, because I was naked; and I hid myself.' יא וַיֹּאמֶר--מִי הִגִּיד לְךָ, כִּי עֵירֹם אָתָּה; הֲמִן-הָעֵץ, אֲשֶׁר צִוִּיתִיךָ לְבִלְתִּי אֲכָל-מִמֶּנּוּ--אָכָלְתָּ. 11 And He said: 'Who told thee that thou wast naked? Hast thou eaten of the tree, whereof I commanded thee that thou shouldest not eat?'

יב וַיֹּאמֶר, הָאָדָם: הָאִשָּׁה אֲשֶׁר נָתַתָּה עִמָּדִי, הִוא נָתְנָה-לִּי מִן-הָעֵץ וָאֹכֵל. 12 And the man said: 'The woman whom Thou gavest to be with me, she gave me of the tree, and I did eat.' יג וַיֹּאמֶר יְהוָה אֱלֹהִים לָאִשָּׁה, מַה-זֹּאת עָשִׂית; וַתֹּאמֶר, הָאִשָּׁה, הַנָּחָשׁ הִשִּׁיאַנִי, וָאֹכֵל. 13 And the LORD God said unto the woman: 'What is this thou hast done?' And the woman said: 'The serpent beguiled me, and I did eat.'

What people knew before the Law was given to the Israelites was more limited than the Law. The Law was given to the Israelites specifically -- for example, the circumcision would be the sign of the Covenant God made with the seed of Abraham. Other nations did not need to do this.

http://en.wikipedia.org/wiki/Seven_Laws_of_Noah are traditionally believed to apply to all humans, but the Law was supposed to be for the Israelites.

Before the Law was given to Israelites, we know that people still knew murder was wrong. For example, http://en.wikipedia.org/wiki/Code_of_Hammurabi dates back to the 17th century BC, and codifies what should be done for what actions.

People of all nations were able to learn morality from their parents, and their human nature was able to understand what would hurt others, as well as what their culture considered acceptable or not. The point of the Law was to maintain a special Covenant between God and His People the Israelites.

The Council of Jesus' Church in Jerusalem officially stated as much, in Acts 15:

https://www.biblegateway.com/passage/?search=Acts+15

They specifically addressed Gentile Believers and told them to follow the Noahide Laws, telling them that they don't have to observe the full Jewish law, which was given to Israelites only.

From a specifically Catholic perspective:

Even though God had not yet given a law before Moses, humans were still called by God, and searched for God:

The dignity of man rests above all on the fact that he is called to communion with God. This invitation to converse with God is addressed to man as soon as he comes into being.

(Gaudium et Spes, Vatican II, paragraph 19. Cited in Catechism of the Catholic Church par. 27)

Further, God responds to this desire for communion with him, and always has, even between the Fall and the Law:

"God, who creates and conserves all things by his Word, provides men with constant evidence of himself in created realities. And furthermore, wishing to open up the way to heavenly salvation, he manifested himself to our first parents from the very beginning." He invited them to intimate communion with himself and clothed them with resplendent grace and justice.

(Catechism, par. 54)

Furthermore,

Deep within his conscience man discovers a law which he has not laid upon himself but which he must obey. Its voice, ever calling him to love and to do what is good and to avoid evil, sounds in his heart at the right moment.... For man has in his heart a law inscribed by God.... His conscience is man’s most secret core and his sanctuary. There he is alone with God whose voice echoes in his depths. ...
"Conscience is a law of the mind; yet [Christians] would not grant that it is nothing more; I mean that it was not a dictate, nor conveyed the notion of responsibility, of duty, of a threat and a promise.... [Conscience] is a messenger of him, who, both in nature and in grace, speaks to us behind a veil, and teaches and rules us by his representatives. Conscience is the aboriginal Vicar of Christ."

(Catechism, parr. 1776, 1778. The quote in the second paragraph is cited "John Henry Cardinal Newman, “Letter to the Duke of Norfolk,” V, in Certain Difficulties felt by Anglicans in Catholic Teaching II (London: Longmans Green, 1885), 248." Emphasis added.)

So, from a Catholic perspective, it's never really been the case that there was no Law; the law was always there, written on our hearts: the Law given by our Maker to all that he made. Compare also Romans 2:14 - 15a (which I see @bruisedreed has also cited):

For when the Gentiles who do not have the law by nature observe the prescriptions of the law, they are a law for themselves even though they do not have the law. They show that the demands of the law are written in their hearts, while their conscience also bears witness ...

(New American Bible: Revised Edition)

This—what Catholicism calls our "moral conscience" was the law before The Law; a law binding on all because it was given to all made by God.