You want a between query based on your date column.
WHERE date BETWEEN startdate AND enddate.
Between is equivalent to date >= startdate AND date <= enddate. It would of course be also possible to just use >= AND < explicitly which would simplify it a bit because you don't need to find the last day of the month, but just the first day of the following month using only DATE_ADD(..., INTERVAL 1 MONTH).
However startdate and enddate in this case would be derived from CURDATE().
You can use CURDATE(), MONTH(), DATE_ADD and STR_TO_DATE to derive the dates you need (1st day of current month, last day of current month). This article solves a similar problem and all the techniques needed are shown in examples that you should be able to adapt:
http://www.gizmola.com/blog/archives/107-Calculate-a-persons-age-in-a-MySQL-query.html
The first day of the current month is obvious YEAR-MONTH(CURDATE())-01. The last day you can calculate by using DATE_ADD to add 1 Month to the first day of the current month, then DATE_ADD -1 Days.
update-
Ok, I went and formulated the full query. Don't think str_to_date is really needed to get the index efficiency but didn't actually check.
SELECT count(*)
FROM entries
WHERE `date` BETWEEN
CONCAT(YEAR(CURDATE()), '-', MONTH(CURDATE()), '-', '01')
AND
DATE_ADD(DATE_ADD(CONCAT(YEAR(CURDATE()), '-', MONTH(CURDATE()), '-', '01'), INTERVAL 1 MONTH), INTERVAL -1 DAY);