Standard numerals are the size of capital, or uppercase, letters. Their best uses are for standalone numbers like your house number, or, because they have a single set width, for columns of tabular material. Because they line up in columns, they are also referred to as lining numerals. Lining numerals are standard in almost every typeface, and are by far the most widely used.
Old-style numerals are the equivalent of lowercase letters — small x-height, plus ascenders and descenders. They’re the best choice to use in a body of text — for dates, times, addresses, phone numbers, and so on, as in, “He lives at 6451 Elm Street,” or, “The 1979 version had 42,362 points.”
Not all typefaces have old-style numerals — which are sometimes called expert numerals — even as an option. The most widely used typeface that has old-style numerals as the default is web favorite Georgia, which you’re now reading.