Lieutenant is the only spelling of the word denoting a second in charge, a deputy, or a rank in the armed forces and (in the United States) police services. The spelling is the same in all varieties of English, regardless of pronunciation. Confusion sometimes arises because, in the U.S., the word is routinely said “lootenant” (or sometimes “lyootenant”), while in the United Kingdom and other countries of the British Commonwealth the preferred pronunciation is “leftenant.” The “American” pronunciation is, however, becoming commonplace in countries like Australia, New Zealand, Canada, and even the U.K., albeit mostly outside official usage.
Like a great many words in English (e.g. drought, colonel, sergeant, debt, etc.), the modern pronunciation may not be phonetic and sometimes seems to be downright antiphonetic. The British pronunciation of lieutenant derives from its history, much of which remains obscure. Premodern spellings (e.g. luff-, leif-, etc.) show that the “lef-” pronunciation has a long history but was by no means the sole one. At some point, spelling and pronunciation diverged in Britain, only to converge again later in the United States. To confuse matters further, the British Royal Navy traditionally pronounced the word “luhtenant,” although this seems to have fallen out of favor.
Neither the British nor the American way of saying lieutenant is inherently better or worse than any other. The choice depends on context (it might be inappropriate to say “lootenant governor” in Canada, for instance) and, to some extent, personal preference.