These three properties are usually known as the CIA triad: confidentiality, integrity and availability. They are the three main classes of security properties of any system (though not all properties fit within those three classes).
Confidentiality (i.e. secrecy) means that an attacker cannot read data that she is not supposed to be able to access. An SQL injection attack is one way to break confidentiality, if the attacker is able to inject or modify a SELECT statement (for example). An SQL injection attack could also compromise integrity, if the attacker can inject an INSERT or an UPDATE or a DELETE or other statements that modify the database.
Integrity means that the attacker cannot write to the database in unauthorized ways. It covers the addition of extra entries, modification of existing entries, deletion of entries, and anything else that changes the data. Using a trojan could compromise the database, but accessing objects with higher security roles would only be a breach of integrity if you meant a write access: a read access would break confidentiality.
Availability means that the attacker cannot prevent others from using the database. A DDOS attack is indeed a way to break availability.
There are, of course, many other ways to break these properties.