Using Joda-Time, take a look at DateTimeFormat; it allows parsing both kind of date strings that you mention (and almost any other arbitrary formats). If your needs are even more complex, try DateTimeFormatterBuilder.
To parse #1:
DateTimeFormatter f = DateTimeFormat.forPattern("yyyy-MM-dd HH:mm:ss");
DateTime dateTime = f.parseDateTime("2012-01-10 23:13:26");
Edit: actually LocalDateTime is a more appropriate type for a datetime without a time zone:
LocalDateTime dateTime = f.parseLocalDateTime("2012-01-10 23:13:26");
And for #2:
DateTimeFormatter f = DateTimeFormat.forPattern("MMMM dd, yyyy");
LocalDate localDate = f.parseLocalDate("January 13, 2012");
And yes, Joda-Time is definitely the way to go, as far as Java date & time handling is concerned. :)
As mostly everyone will agree, Joda is an exceptionally user-friendly library. For example, I had never done this kind of parsing with Joda before, but it took me just a few minutes to figure it out from the API and write it.
Update (2015)
If you're on Java 8, in most cases you should simply use java.time
instead of Joda-Time. It contains pretty much all the good stuff—or their equivalents—from Joda. For those already familiar with Joda APIs, Stephen Colebourne's Joda-Time to java.time migration guide comes in handy.
Here are java.time versions of above examples.
To parse #1:
DateTimeFormatter f = DateTimeFormatter.ofPattern("yyyy-MM-dd HH:mm:ss");
LocalDateTime dateTime = LocalDateTime.from(f.parse("2012-01-10 23:13:26"));
You cannot parse this into ZonedDateTime or OffsetDateTime (which are counterparts of Joda's DateTime, used in my original answer), but that kinda makes sense because there's no time zone information in the parsed string.
To parse #2:
DateTimeFormatter f = DateTimeFormatter.ofPattern("MMMM dd, yyyy");
LocalDate localDate = LocalDate.from(f.parse("January 13, 2012"));
Here LocalDate is the most appropriate type to parse into (just like with Joda-Time).