The sentence that that I heard was 頼りにしている。 Google Translate tells me that means "I'm counting on you". My understanding of Japanese tells me "this doesn't compute". Is there a general rule for translating にしている or している in sentences like this? What's the general meaning of that phrase?
頼り is the noun form of verb 頼る (tayor-u), and here 頼り means a thing/person to count on. Examples:
君だけが頼りだ。 I can only count on you.
地図を頼りに家を探す look for the house with the help of a map (This example is from Daijisen; the English translation is by me.)
In general, AをBにする means “turn A into B.” Setting B=頼り, Aを頼りにする means “turn A into ‘something to count on’,” or in more natural English, it means “count on A.”
(Well, strictly speaking, Aを頼りにする does not turn anything into something different, but AをBにする is used even in this case. Another example of this usage is 遠足を楽しみにする (look forward to the picnic); see Daijirin, sense 【2】.)
In your case, A is not specified. I am a little bit fuzzy here, but I think that by default, 頼りにしている means あなたを頼りにしている. Therefore, Google Translate is correct in this case: it means “I am counting on you.”
To start with, 頼りにしている does translate as "(I'm) counting on you" or "(I'm) relying on you".
Basically in this case it's saying that the speaker is in this situation of relying on someone for something. "I am relying on you to bring back my library book, because otherwise I'll get a fine". It's describing the speakers state.
The second point I'd like to make is that it might be easier for you to break up 頼りにしている into components. 頼りにする is it's own unit (頼む==>ます-stem + に + する). TsuyoshiIto has explained it really well.
The other grammar point is ～ている, which is really handy to know know and learn, and it has various meanings depending on how it's used. It's basically the て-form, plus いる which is "to be" (for living things).
In general there are two uses to describe one's state for the ～ている grammar point. One that describes something that is happening now (drinking tea), and the other describes something that happened in the past, and is still true in the present (being married).
今はお茶を飲んでいます。 (I'm) drinking tea right now.
田中さんは結婚しています。 Tanaka is married.
This could get a little ambiguous sometimes in a sentence like this:
私は日本語を勉強しています。 I'm studying Japanese (right now) / I study Japanese (at school)
The third use for ～ている is to describe something habitual.
毎日、新聞を読んでいます。 I read the paper every day.
Now, I'll leave you with a bunch of links for more information on ～ている: