The examples below are sometimes encountered in scientific contexts and ordered from least to most challenging to my indulgence.

10. latter vs. later. Latter refers to the last thing mentioned. Later means "at a subsequent time".

9. uniqueness vs. unicity. In mathematics, uniqueness is the quality of something being unique. Unicity may sound natural for some foreign-born people but should generally be avoided in that context. (I do not dispute that there may be obscure, specialized mathematical concepts like strong unicity in the literature but I think it would be unfair to use them as an excuse.)

8. discreet vs. discrete. Discreet essentially means quiet. Discrete is the quality of something not being continuous, like the integers in mathematics.

7. complementary vs. complimentary. Complimentary means free; a complementary thing serves as a complement. I'm always suspicious of "complementary rice" on restaurant menus.

6. stationary vs. stationery. Something stationary does not change in time; stationery refers to office supplies like paper and envelopes.

5. recieve (receive). For years I lived near near an auto body shop with a sign that said "recieving parts". Trauma ensued.

4. definately (definitely). "Definitely" makes your point stronger. "Definately" makes it a whole lot weaker.

3. affect vs. effect. In short, "affect" is most often used as a verb and "effect" is most often used as a noun. A affects B if A has an influence on B. An effect is a result; something that follows from a cause. Note that "affect" is also a noun that means feeling or emotion in some rare contexts; and "effect" is also used as a verb: A effects B if A produces B as a result. The last two uses are however much less common.

2. principal vs. principle. Principal is mostly used as an adjective to mean the most important, like in "principal components analysis", but also as a noun, for instance in financial situations. A principle is a basic truth, or a fixed set of rules, like the "uncertainty principle".

1. its vs. it's. This one gets a medal. I've even seen the New York Times misuse "it's" more than once. "Its" indicates possession by an object, and is the analogue of the pronouns his or her. "It's" is the contraction of "it is". Some related homonyms involving two or three entirely different grammatical entities are: their/they're/there, we're/were/where, your/you're, who's/whose, should of/should've, etc. On a related note, an apostrophe followed by s never indicates the plural form; nobody has "two car's".