Dynamic and formal equivalence are two methods or styles used to convert source text (e.g. Hebrew or Greek) into another language (e.g. English).
The Dynamic (also known as functional) method attempts to convey the thought expressed in the source text using equivalent expressions from a contemporary language like English ('thought for thought' translating).
The formal equivalence method (also known as a literal translation) attempts to translate the source text word for word into another language.
Differences
The difference between these translation techniques can be seen by comparing the first few words of Genesis 1 in the Good News Translation (dynamic) to the American Standard Version (formal). Also included is a Bible that uses a mix of both translation techniques (Holman Christian Standard Bible).
In the beginning, when God created the universe, the earth was formless and desolate (GNT).
In the beginning God created the heavens and the earth. And the earth was waste and void; (ASV).
In the beginning God created the heavens and the earth. Now the earth was formless and empty (HCSB).
Bibles that utilize a Dynamic translation technique can be good for conveying the overall meaning of a passage. Those that use this method include the Contemporary English Version (CEV), Good News Bible (GNB), New English Bible (NEB) and New Jerusalem Bible (NJB).
Bibles that utilize a formal (functional) technique, because of their exactness in translation, are best for detailed, in-depth studies and for research to determine doctrine. Those that use this method include the American Standard Version Bible (ASV), Holy Bible in Its Original Order - A Faithful Version (HBFV), King James Version (KJV), New King James Bible (NKJV), New American Standard (NASB), Young's Translation (YLT) and the New Revised Standard Version (NRSV).
Versions of God's word that use a mix of both Dynamic and Formal Equivalence include the popular New International Version (NIV) and the Holman Christian Standard Bible (HCSB). There are also versions of Scripture that are not, strictly speaking, translations. They are paraphrases of God's word. One example of this type of version is the Living Bible.