When うち, (sometimes seen using the kanji 内（うち）, or 裡（うち）) is paired with the 格助詞（かくじょし） (case marking particle) に, it expresses a period/space ‘within’ which something else will/should happen/exist. It quite often (but not always) carries the nuance of (A) being the ‘best’, or the ‘optimal’ time/space within which (B) could occur/exist. In English, うちに can simply be translated as ‘while’, or ‘during’.
うちに may be paired with the ている form of verbs, the plain form of い-Adjectives, nouns followed by の, or な-Adjectives that are followed by な.

• 彼（かれ）と毎日（まいにち）会（あ）っているうちに、だんだんと彼（かれ）の事（こと）が好（す）きになってきた。
  As I've been meeting with him everyday, I have gradually come to like him. (As I spend more time with him every day, I have grown to like him)
• 熱（あつ）いうちに食（た）べて！
  Eat it while it's hot!
• お爺（じい）ちゃんがまだ元気（げんき）なうちに家族（かぞく）皆（みんな）で旅行（りょこう）に行（い）こう！
  Let's all go on a family trip while grandpa is still healthy!
• 今（いま）のうちに明日（あした）の準備（じゅんび）をしておこう。
  Let's get ready for tomorrow while we have time. (Before it is too late)

Unlike 間（あいだ）に, うちに expresses something that can happen/exist anywhere ‘within’ the span of (A), and is therefore non-specific as to where (B) is located.

• 運転（うんてん）している間（あいだ）に眠（ねむ）くなったので、コンビニで休憩（きゅうけい）をした。
  I got sleepy while driving, so I took a break at a convenience store. (While in the process of driving)
• 運転（うんてん）しているうちに眠（ねむ）くなったので、コンビニで休憩（きゅうけい）をした。
  I got sleepy while driving, so I took a break at a convenience store. (At one point while driving)