As your English ability becomes more advanced, you will want an English-English dictionary. There are two main types: learners' dictionaries and regular dictionaries.
Learners dictionaries explain words in rather easy English and focus on the differences between words that are similar in meaning. They are very useful to people who plan to write a lot in English because they give detailed grammatical information about how the word is used. My favorite learners' dictionary is the Longman Dictionary of Contemporary English; I often use it for linguistic research and in preparing for teaching my classes. There are many other learner's dictionaries, including the Collins Cobuild Dictionary for Learners of English and The Oxford Advanced Learners' Dictionary); look at several and choose the one you like the best.
If you will be doing a lot of reading, you will probably want a regular English dictionary because bilingual dictionaries or learners' dictionaries may not have enough words. These dictionaries usually also have information about famous people and geographical locations, included either in the regular listings or in a separate appendix at the back of the book.
I like the various dictionaries published by Webster (for example, Webster's New World Dictionary and Webster's Collegiate Dictionary) because I almost always find the words I want and because they have etymological information (information about the word's history). Another good feature is that they are available in paperback.
These days, English-English dictionaries are often included as one of the volumes in electronic dictionaries, and some of them are also available on-line. You'll still probably one at least one actual book dictionary in your house too, though.