Most of the answers were good, but a few were misleading on one point: the Roman Catholic Church **does** officially approve some English Bibles (and not others), and the approval includes the advice that the Bible is authorized for use by Roman Catholics.
Here is a list of Bibles that include the Roman Catholic Scriptures. Also shown is the degree of formal approval that they have received from that church (if any). They are listed roughly in order - those with the highest degree of approval at the top, no approval at the bottom.
http://www.bibleselector.com/rc_versions.html
To answer your question directly: the NRSV-CE has been approved for use by individuals but rejected for use in liturgy. The NRSV without the "CE" (= Catholic Edition) has not been approved. If you get an NRSV with Apocrypha, it should have all of the Scriptures as the NRSV-CE but not in standard Roman Catholic order and also a couple of extra books not approved for Roman Catholic Bibles.
Final note: the NRSV is a pretty good Bible - but has some notable shortcomings. If you already have one (either CE or with Apocrypha), I recommend that you go ahead and use it. If you have an edition without the CE and without the Apocrypha you can certainly still use it - but from the Roman Catholic perspective it is missing some of the Bible books and you will want a "complete" Roman Catholic Bible.
Jim
http://www.bible-reviews.com/topics_rc_vs_pr.html#Intro
http://www.bibleselector.com/rc_approval.html
http://www.bibleselector.com/