What does "only" mean?
Solution 1:
The terms "only" and "only if" mean different things. The first is English and the second is mathematical.
1) "only" in English means "just this and not others", where "others" could be an object or an action, depending on what "only" is connected with. For example, "We only proved that $p$ is prime" means you showed $p$ is prime but not anything further, which I think is synonymous with "We proved only that $p$ is prime"; it is just a stylistic judgment as to which of those you use and I prefer the second version at the moment. However, "We proved that only $p$ is prime" means you proved $p$ is prime while other numbers in the argument (that may possibly have been prime) are not prime.
The classical example, which is not meant as a personal remark, is to insert "only" in front of each word of the sentence "I love you". Each version has a completely different meaning. This is discussed under the topic "Modifiers" on the page http://www.kaptest.com/GMAT/Learn-and-Discuss/Community/blogs/tag/verbal/
2) In math, "only if" has the same meaning as "implies". (Logically, it's the direction that is not "if", which is how I first was able to remember its meaning.) In terms of Tom's example, "shoes are on ==> socks are on" has the same meaning as "your shoes are on only if your socks are on."
Solution 2:
I think analogies in plain English are the way to internalize this... so here's one:
Given that you want to wear socks with your shoes, put your shoes on only if you have already put your socks on.
The idea is that there is no other way to arrive at the state of having your socks and shoes on (aside from the ridiculous possibility of placing your socks over your shoes).
Solution 3:
Here are some links from other websites which helped me:
1.http://courses.umass.edu/phil110-gmh/text/c04_3-99.pdf
2. https://english.stackexchange.com/questions/32343/if-vs-only-if-vs-if-and-only-if
3. http://aristotle.tamu.edu/~rasmith/Courses/Logic/Exercises/1.3.extra.pdf
4. http://forums.philosophyforums.com/threads/only-if-vs-if-and-only-if-27842.html