The British form written with digits is 3/10/10 or 3.10.10 (or the year may be given in full) and means 3 October 2010. When we read it, we say, "the third of October, two thousand (and) ten." Notce that the words "the" and "of" are not included in the written form or running text.
The American form is 3/10/10 or 3/10/2010 and could also be written March 10, 2010. When we read it, we say, "March (the) tenth, two thousand ten." Notice that the month and day are exactly opposite to the British form.
A comma must separate the named day and the year in the American style. There is no comma between the month and year in the British style. However, if you add the day of the week, you must separate it with a comma, e.g., Tuesday, 19 June 2010.
Only when copying an older source is it correct to write ordinal numbers for dates, i.e., with the endings -st,-rd, or -th showing the day (wrong: 12th April 2010). However, we can use the ordinal form in an incomplete reference (The conference dates were 4-6 May 2010 but most people stayed until the 7th).
Do not use the digit form of dates in running text (8/2/10). You should also write out the month if used alone or without the month, date and year, e.g., We met in January, on 12 January to be exact. (not Jan. or 12 Jan.).