あるいは is a construction in Japanese that is used as a 接続語（せつぞくご） (conjunction). It usually appears after statements that have been marked with the 副助詞（ふくじょし） (adverbial particle) か. あるいは may be loosely translated as ‘possibly’, ‘maybe, or ‘or’.

• 運動（うんどう）をさせるかあるいは餌（えさ）を減（へ）らしてください。
  Please make them exercise or possibly lessen the amount of food.
• パソコンの動作（どうさ）が遅（おそ）いかあるいは電源（でんげん）が点（つ）かなくなった場合（ばあい）は、私達（わたしたち）が直（なお）します。
  In the event that the computer gets slow or doesn't turn on, we will fix it.
• 有名（ゆうめい）かあるいは綺麗（きれい）であれば誰（だれ）でも入（はい）れるらしい。
  I heard that if you are famous or maybe beautiful, anyone can get in.
• 土曜日（どようび）かあるいは日曜日（にちようび）までには終（お）わらせておきます。
  I will try to have it finished by either Saturday or possibly Sunday.

Although usually being treated as a single word, あるいは is actually a combination of the 連体形（れんたいけい） (attributive form) of 或（あ）る, a verb meaning ‘a certain’, the 接尾辞（せつびじ） (suffix character) い (used for making い-Adjectives), and the 副助詞（ふくじょし） (adverbial particle) は. Due to this, the literal meaning of あるいは is just ‘another possibility’, as it highlights a certain something existing within the same area/category as (A).

• 牛乳（ぎゅうにゅう）あるいはチーズが使（つか）われている料理（りょうり）は食（た）べられません。
  I can't eat anything that uses milk, or another possibility is cheese.
• 月曜日（げつようび）あるいは火曜日（かようび）に新（あたら）しい生徒（せいと）が来（き）ます。
  A new student will come on either Monday, or another possibility is Tuesday.