You can search in the Korean box using Korean hangeul, in the English box using English, and in the Romanisation box using revised romanisation. With English, you can ignore the capitalised first letter that Android insists on supplying -- the search will cover both large- and small-case. With romanisation, you currently have to use hyphens to separate the syllables as written in jamo, eg "neo-mu" instead of "neomu" -- that may change in the future.
The results are returned in two groups -- the first contains exact matches for your search, and the second contains matches which contain your search.
The results consist of: the Korean in hangeul, the part of speech in brackets, a representation of the hangeul in revised romanisation, the English equivalent(s) of the Korean, and any citations for the result in (an older version of) Tatoeba.
Currently, you can only search for single words, not phrases. This will probably get addressed later.
The textfile icon to the right of the Kosajeon title will open a list of affixes. Click on an affix and you can see all currently-listed affixes, with general meanings.