The Julian date (Julian day, JD) is the number of days since a chosen fixed time, specifically noon, UT, November 24, 4714 BC Gregorian calendar, (noon January 1, 4713 BC Julian calendar). It is of note that Julian dates are noon-to-noon rather than midnight-to-midnight, i.e., date 2451545.0 represents what we would call noon UT of some day and 2451544.9583 (i.e., 2451545.0 minus 1/24) is what we would call 11AM UT of the same day.
MHJD can mean "modified HJD" as above, or can be used for a version adjusted by some other constant. Analogously for MBJD.
A Julian year is somewhat distinct from Julian dates and the Julian calendar: it is exactly 365.25 days of 86400 SI seconds. It is used in astronomy as a measure of time.