So 或者 and 还是 both suggests offering options, and are usually translated as “or” in English. The difference is that, 或者 is offering options in a declarative sentence/a statement to offer options, while 还是 is usually offering options to ask a question.

We'll learn about 或者 first.

To use 或者 in a declarative sentence, we can use this structure, Option 1, Opition 2 or more, and put 或者 before the last Option (X), following this structure:

Option¹……，Option²……，……，或者 Option^X

For example,

wǒ xiǎng xué chànggē, tiàowǔ, huòzhě huàhuà
我想学唱歌，跳舞，或者画画。
I want to learn to sing-song, dance-dance, or paint-paint.
wǒmen kěyǐ kàn Shànghǎi duì de bǐsài, huòzhě kàn Běijīng duì de bǐsài
我们可以看上海队的比赛，或者看北京队的比赛。
We can watch Team Shanghai's game, or watch Team Beijing's Team.

Grammar 2: "Or" in Chinese with 还是 · HSK 2

To ask a question that offers options for people to choose from, we can use 还是 instead of 或者. Questions like this are called Alternative Questions, 选择疑问句(Xuǎnzé Yíwèn Jù). To ask a question with 还是, there can be two structures.

(1) Option¹……，Option²……，……，还是 Option^X？

(2) 是 Option¹……，Option²……，……，还是 Option^X？

For example,

nǐ (shì) xǐhuan yīntiān, qíngtiān, háishì yǔtiān
你喜欢阴天，晴天，还是雨天？
你是喜欢阴天，晴天，还是雨天？
(Do) you like cloudy days, sunny days, or rainy days?
Lǎo Zhāng (shì) wǔyuè jiéhūn, liùyuè jiéhūn, háishì qīyuè jiéhūn
老张五月结婚，六月结婚，还是七月结婚？
老是张五月结婚，六月结婚，还是七月结婚？
(Does) Old Zhang get married (in) May, June, or July?

Be careful that sometimes 或者 can be used in a question, but it is not used there as the structure that raises the question. For example,

chànggē, tiàowǔ, huòzhě yóuyǒng, nǐ xiǎng xué nǎge
唱歌，跳舞，或者游泳，你想学哪个？
Sing-song, dance-dance, or swim-swim, which one (do) you want to learn?
lánqiú huòzhě zúqiú ba
篮球或者足球吧？
Basketball or soccer maybe?

In the above sentences, it is 哪个 and 吧 that is used as the structure that raised the question, not 或者.