Is there a terminological difference between "sequence" and "complex" in homology theory Suppose you are given something like this:
$\dots \longrightarrow A^n \longrightarrow A^{n+1} \longrightarrow \dots$
People tend to talk about "chain complexes" but about "short exact sequences". Is there any terminological difference or any convention with regards to using these words (EDIT: I mean "complex" and "sequence" in a homological context) that a mathematical writer should comply to?
 A: A small remark: the sequence $...\rightarrow A_n\rightarrow A_{n+1}\rightarrow A_{n+2}\rightarrow A_{n+3}\rightarrow ...$ can be "exact" or a "complex" (it depends on the nature of the maps, as confirmed in the answer by @exitingcorpse), but not a "short exact sequence".
Short exact sequences (of $R$-modules, for example, denoting by $R$ a given ring) are exact sequences of the form  $0\rightarrow A\stackrel{\alpha}{\rightarrow} B\stackrel{\beta}{\rightarrow} C \rightarrow 0$; from the definition of exact sequence plus the shortness condition it follows that $ker(\beta)=im(\alpha)$, and $\alpha$ is injective and $\beta$ is surjective.
A: To say that the sequence is a chain complex is a less imposing condition: it simply says that if you compose any two of the maps in the sequence, you get $0$. But to say the sequence is exact says more: it says this is precisely (or, if you rather, exactly) the only way you get something mapping to zero. 
The first statement says that the image of the arrow to the left of $A^n$ is contained in the kernel of the arrow to its right, the second statement says that the reverse inclusion also holds.
A: Let $R$ be a ring. A chain complex is a sequence (of left $R$-modules)
$A = \cdots\to A_{n}\xrightarrow{d} A_{n-1}\xrightarrow{d}\cdots$
Such that $d^2 = 0$. This allows you to define $H_n(A) = \ker{d_n}/\mathrm{im}~d_{n+1}$.
An exact complex is a chain complex such that $H_n(A) = 0$ for all $n$. Exact complexes are usually called acyclic complexes.
A short exact sequence is an exact chain complex of the form $0\to A\to B\to C\to 0$. This is equivalent to saying $A\to B$ is an injection, and $B\to C$ is a surjection whose kernel is the image of $A$ under $A\to B$.
