Argument of align has an extra '}'

The expression

 f(t) = ((x_t - x_{t-1})}

has one opening brace and two closing braces and that triggering the error message. The proper way to write the expression would be

f(t) = ((x_t - x_{t-1}) + (x_{t-1} - x_{t-2}))

or even better

f(t) = \bigl((x_t - x_{t-1}) + (x_{t-1} - x_{t-2})\bigr)

so the outer parentheses will be longer and this will increase readability.

On a side note, as egreg has mentioned, your code has also some other problems: the center environment is not necessary here and will add extra unwanted vertical space. For a single equation you should use equation instead of align. You shouldn't use two consecutive align environments; you can use a gather environment instead.

It's not clear why you want to split in that way a single formula; the breaking point that you've chosen and the numbering schema introduce ambiguity since the reader could get the impression that you are talking about two different expressions instead of two parts of a same formula. Perhaps you culd consider using \underbraces? Here are some options:

\documentclass{article}
\usepackage{amsmath}

\begin{document}

\begin{gather}
    \label{avggrad}
    f(t) = \big((x_t - x_{t-1}) + (x_{t-1} - x_{t-2})\big) \times \frac{1}{2} \\ 
    \label{avggradsum}
  \frac{ \sum_{i=0}^{n-1} (x_{t-i}-x_{t-(i+1)}) }{n}
\end{gather}

\begin{equation}
    \label{avggrad}
    f(t) = \big((x_t - x_{t-1}) + (x_{t-1} - x_{t-2})\big) \times \frac{1}{2} \frac{ \sum_{i=0}^{n-1} (x_{t-i}-x_{t-(i+1)}) }{n}
\end{equation}

\begin{equation}
    \label{avggrad}
    f(t) = \underbrace{\big((x_t - x_{t-1}) + (x_{t-1} - x_{t-2})\big)}_{\text{some explanation}} \times \frac{1}{2} \underbrace{\frac{ \sum_{i=0}^{n-1} (x_{t-i}-x_{t-(i+1)}) }{n}}_{\text{some description}}
\end{equation}

\end{document}

The "correct" input is

\begin{gather}
    \label{avggrad}
    f(t) = ((x_t - x_{t-1}) + (x_{t-1} - x_{t-2})) \times \frac{1}{2}\\
    \label{avggradsum}
    \frac{ \sum\limits_{i=0}^{n-1} (x_{t-i}-x_{t-(i+1)}) }{n}
\end{gather}

Don't use align for single equations and never have two consecutive equation alignment environments.