Abbreviations, contractions, and acronyms are ways of shortening a word or phrase but sometimes can be confusing for a novice writer. I will take each one briefly in turn.
Abbreviations are defined by the Oxford English Dictionary (OED) as “a shortened form of a word or phrase”. These are often formed by omitting the end of a word or words, such as in assoc. (association) or co. (company, usually in the business sense). Notice that the abbreviations here end with a full stop (or period) in British English.However, in US English full stops are often used where the abbreviation or contraction ends in a lower-case letter as in Dr. or Ms. as recommended by the Chicago Manual of Style. In contrast, in upper case abbreviations such as CEO, UK, and US, the full stop is often omitted.
Of course, as with most rules there are exceptions in US English, such as PhD (a mix of upper- and lower-case letters). Modern US usage is for CA, NY, and other state names to be without full stops, but some more traditional publications still use them as in N.Y., and in the country abbreviations of U.S. and U.K.Lower case abbreviations such as e.g., i.e., and pp. normally use full stops, but there is a growing trend for these to be omitted, as in mph (miles per hour) and especially in scientific texts.
The OED defines a contraction as a “word or group of words resulting from shortening and original form: goodbye is a contraction of “God be with you”. These can be formed by omitting the middle of a word or words, such as in Dr (doctor), Ltd (Limited, again usually in the business sense, i.e. Limited Company in the title of a business), and St (for Saint or Street). In UK English word contractions usually do not take a full stop. In the example of Smith & Jones Co. Ltd the Co. has a full stop as it is an abbreviation whereas Ltd does not require a full stop as it ends with the last letter that would appear if spelt out in full. Note that plc does not end in a full stop and can appear as lower- or upper-case initials.
Contractions also appear in everyday speech: can’t, it’s, I’ve, and mustn’t. Apostrophes are used to indicate where letters have been omitted to form the contraction. Sometimes, in more formal publications such as academic works, such contracted words would not be permitted, and the words would be spelt out in full, such as cannot, it is, I have, and must not.
An acronym is an “an abbreviation formed from the initial letters of other words and pronounced as a word” (OED) such as NATO or NASA. Similar to these are initialisms where the initial letters of words are used but are not usually pronounced as one word, as in BA, BBC, FBI, QC, and SE. Full stops are not used after each letter in UK English. It can also be applied to phrases, such as OMG (oh my God!).
For guidance on the above areas refer to the New Oxford Guide Style Manual for UK English or The Chicago Manual of Style for US English.