Thank you for your question.
"Riba" means "interest" as it is used in finance terms. Basically, what the prohibition amounts to is that no contract may involve the exchange of similar commodities or money where one party gets an increase. So, for example, if the contract states "you give me 5 pounds and I will give you 5 pounds back plus an increase of 1 pound" it is not allowed.
Here are some references to explain the meaning of Riba. First in the Qur'an:
O ye who believe! Devour not usury, doubled and multiplied; but fear God; that ye may (really) prosper.
Surah 3 Verse 130
As for hadith, there are several hadith on the subject which you can find on the internet. In particular,
"Gold for gold either ore or pure, silver for silver either ore or pure, wheat for wheat measure for measure, barley for barley measure for measure, dates for dates measure for measure, salt for salt measure for measure whoever increases or seeks an increase has committed riba. There is nothing wrong with selling gold for silver and the silver is more as long as it is hand to hand, as for deferred payment, no. And there is nothing wrong with selling wheat for barley and the barley is more as long as it is hand to hand, as for deferred payment, no." In another version, he (peace be upon him) said: "When the items are different in these categories, then sell however you wish as long as it is hand to hand."
Abu Daud and both narrations are sahih.
Some people try to explain this in various ways dealing with effortless profit or taking risk. However, these explanations really are unnecessary. The rule is quite clear. One way that is helpful in understanding the issue is to take a contract that looks a lot like riba but which is allowed in Islam and compare it to one that is not allowed:
Let us suppose I am a farmer, and I need to harvest my crop, but I don't have enough money to pay for the all the costs. One way to finance the extra costs is to take out a loan with interest. Say, borrowing 1000 pounds for six months and paying back 1050 pounds. This however, is not allowed in Islam. Another way to finance the extra costs would be to buy what is needed with a contract specifying delayed payment. Say, 1000 pounds worth of goods to be paid for after 6 months at the price of 1050.
On the surface both arrangements seem almost identical.
The key difference is that in the contract where there is delayed payment for goods, the use of the money is apparent. This matters when there arises a difficulty in meeting the terms of the contract. In such cases, a judge may be able look at the use of the money and order an appropriate and just resolution. Enforcing a contract that does not specify the use of the money allows not only completely illegal use of loaned money, but also enforcement of payment against all principles of justice.
We see this brutal and unjust enforcement in many cases, throughout history, and recently paying interest on debt has brought many countries to their knees, with interest payments taking top most priority over fundamental life or death issues such as funding essential healthcare, education or paying pensions. On an individual level allowing interest based contracts can be used to force people into the direst of circumstances which can and does lead to their deaths.
If riba is banned, on the other hand, then the law and justice rules over the amoral interests of money.