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For some writers, a coordinator (and, but, so, or, yet) is only used between two independent clauses (separated by a comma.) That is, a "conjunction" is not used at the beginning of a sentence. Grammarians (scholars in language use) do not support this restricted use.  (See comments below.)

For other writers, including journalists and academic writers, a coordinator or "conjunction" is used both between two clauses and at the beginning of a sentence.  Read the comments below from well-known grammarians. 

In traditional grammar, a conjunction and a coordinating conjunction differ in that a conjunction joins grammatically alike subclausal elements, whereas a coordinating conjunction joins grammatically alike clausal elements. (Azar )  (PEU 510.1-2)

In linguistic description, "coordination is the relation between elements of equal syntactic status, and as such contrast with subordination [unequal syntactic structure]."    (CaGEL  "Properties of prototypical coordinators"  15 §2.1)   

conjunction – is a term for a word that joins two like elements, such as nouns, verbs, adjectives, adverbs, or phrases (but not clauses).

conjunction – is a logic function of A + B "both";   ("and" and sometimes "or") Caffeine is found in coffee and tea. Caffeine is found in coffee or tea. ("both").
disjunction – is the logic function of  A / B "one or the other, but not both", "either"; ("or" / "nor") Would you like coffee or tea?  ("one not the other")
 

coordinating conjunctions —for, and, nor, but, or, yet, and so — join independent clauses.  The resulting construction is a compound sentence.

coordinators — and, but, or, and nor— join a variety of syntactically alike structures both subclausal (NP, VP, AdjP, Adv, etc.P) and clausal. 

symmetric constructions –  He got up and he poured some coffee.  (Clause + Clause)
asymmetric constructions – He got up and  __ poured some coffee. (Clause + NP)   (CaGEL "gapped coordination" 15 §4.2);

conditional constructions with and – Look at it and you'll see what I mean.  (CaGEL 15 §2.2.2.4)
 

yet and so – In traditional grammar, yet and so are coordinating conjunctions. In linguistic description, they are connective adverbs, but they may be regarded as "marginal members of the coordinator category" when used in combination with a coordinator:  He took a taxi, and so he arrived early.  He took a taxi, and yet he arrived late. (CaGEL 15 §2.10)

connective adverbs — so (result) and yet (contrary to expectation)— join a subordinate clause which may be of unequal syntactic status; do not allow reversible clause position; and may be fronted.   (CaGEL 15 §2.10)  

for, only and so+that  are borderline connectors / prepositions (which take clausal complements)  Huddleston and Pullum (CaGEL)  call them "problematic".   He called a taxi, for he wanted to go home; He wanted to call a taxi, only (except) he had no phone number; He called a taxi, so that he could go home.  (CaGEL 15 §2.11)

compound sentence– In traditional grammar, the joining of two independent clauses is called a compound sentence.

In current linguistic description, "compound" is limited to the joining of two words: sweetheart, blackbird, copycat, egghead, socio-economic, etc.)  See  Hyphens  compounds.