The Interactive Calendar converts between some of the most important calendars used in Middle-earth and Númenor, which is simple enough to do, since Tolkien gives us all the points of comparison we need. It also converts those dates forward onto a modern (that is, a Gregorian) calendar, which is a little less straightforward. Tolkien himself doesn't directly explain how to make that adjustment, so converting to modern dates involves a certain degree of interpretation and interpolation from the available references. This article discusses how we calculate the conversion on this site, and explains some of the rationale behind that calculation.
There are four main references that give us clues about how to proceed here:
 '...the middle-day (183rd) was called loëndë
 'It appears ... that Mid-year's Day
was intended to correspond as nearly as possible to the summer solstice.'
 'In that case the Shire
dates were actually in advance of ours by some ten days...'
 '...and our New Year's Day corresponded more or less to the Shire
All from The Lord of the Rings Appendix D The Calendars
(Note that ,  and  follow one another directly in the text of Appendix D; we've subdivided them here for ease of reference in the comments that follow.)
On the face of it, then, the process is simple. Following , we just need to match up modern 1 January with 9 January in the Shire, and we have our conversion; but that simply doesn't work. First, the difference between 1 and 9 January is eight days, not ten, so this contradicts . Second, if we follow this through, Mid-year's Day falls on 23 June, too late for the summer solstice, so this contradicts  as well. It seems that when Tolkien says 'some ten days' and 'more or less to the Shire January 9', he really did mean that those values weren't to be taken as exact.
Another approach would be to use  and  in combination, and match up the calendars so that the summer solstice falls on the 183rd day of the year. This is made problematic by the fact that the actual date of the solstice isn't fixed; it's typically on 21 June, but it can vary around that date (for instance, in 2012 it actually fell on 20 June). What's more, if we take the most common date of 21 June and align the calendar based on that, then modern 1 January now corresponds to Shire 11 January, two days too late by comparison with what we're told in .
These two approaches give us calendars that differ from one another by two days, so an obvious possibility would be to take a compromise approach, choosing a point of comparison half way between the two. That would resolve most of the difficulties by bringing all the disparate dates within one day of each other, but at first glance, it seems a rather arbitrary solution. Is there any real justification for this compromise?
The key fact to remember here is that the summer solstice shifts over time. At the beginning of the twenty-first century, it's usually on 21 June, but at about the time Tolkien was writing Appendix D, 22 June was not unusual (in fact the solstice fell on that date in both 1946 and 1947). Unfortunately, we don't know exactly when Appendix D was written; Christopher Tolkien's best estimate comes from volume 12 of The History of Middle-earth:
'...certainly before the summer of 1950 ... and probably earlier: in fact, an envelope associated with D 1 [the earliest manuscript] is postmarked August 1949.'
The History of Middle-earth volume 12
(The Peoples of Middle-earth)
IV The Calendars
Even if we don't know exactly when Tolkien was writing, it's still true say that 22 June was certainly not an incorrect date for the summer solstice in the relevant period. Allowing that as a point of comparison, things start to fit together fairly well, matching all the main references within at most a single day:
- Mid-year's Day falls on 22 June, which is a reasonable match for the summer solstice (and certainly would have been in the 1940's)
- Our New Year's Day would fall on Shire January 10: not an exact match, but 'more or less' January 9, and with 'some' ten days' difference (nine days' difference, to be precise).
It does not seem to be possible to create a perfect correlation between the calendars of Middle-earth and the one we use today, but this compromise solution does seem to be the best fit based all the information we have. Of course, this certainly doesn't mean that it is definitively correct! It's quite possible to produce viable alternatives, but the system discussed here is the basis of the conversion used by the Encyclopedia's Interactive Calendar.