Before looking at "new" let's define covenant. A covenant, in a Hebraic context, is an agreement to have a special relationship. And so there are many types of covenants just as there are many types of relationships (e.g. marital, communal, political, etc).
While the Tanak does at times use berit (covenant) synonymously with Torah (Law), it is important to understand that the two terms have different meanings. This helps us understand passages like Jeremiah 31:31-34. The New Covenant is not a New Torah; rather, it is a completely new agreement in which the already special relationship between G-d and Israel becomes more intimate than ever before. The Law, meanwhile, remains virtually the same (except for authorized improvements to the priestly system).
Okay, now for something a little more advanced. This is McKee explaining that "b'rit chadashah" (Hb., "New Covenant") and diatheken dainen (Gk., "New Covenant") do not mean "Renewed Covenant":
"The most common Hebrew term used in the Tanach for 'new' is the verb chadash... In the Piel stem (intensive action, active voice) it can mean 'to make anew, restore' (HALOT). It is employed in 2 Chronicles 15:8 as such when King Asa 'renewed [chadash] the Altar of HASHEM that was before the Hall of HASHEM' (ATS). However, the adjective chadash...does not have the same variance that its verb equivalent has. It is used to indicate things that are 'new, fresh...not yet existing,' 'new things' (HALOT). In this way the b'rit chadashah of Jeremiah 31:31-33 is to truly be a New Covenant that is unparalleled by what has come before it..." pg. 138 of Hebrews For the Practical Messianic.