In the absence of any style keys for the \addplot commands, the default cycle list (color, as documented in Section 4.7.7 of the v1.11 pgfplots manual) is used:
\documentclass{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.11}
\begin{document}
\begin{tikzpicture}
\begin{axis}[domain=0:1]
\addplot {x};
\addplot {x^2};
\addplot {x^3};
\addplot {x^4};
\end{axis}
\end{tikzpicture}
\end{document}

As soon as any style keys are specified, as in \addplot[red] <...>;, the default cycle list is no longer used for that plot (but observe that the plot's "spot" in the cycle list is still "consumed", as shown in the example):
\documentclass{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.11}
\begin{document}
\begin{tikzpicture}
\begin{axis}[domain=0:1]
\addplot[red] {x}; % added [red] here
\addplot {x^2};
\addplot {x^3};
\addplot {x^4};
\end{axis}
\end{tikzpicture}
\end{document}

The alternative plot command \addplot+ allows the user to append style options without overriding all styles set by the cycle list. For example:
\documentclass{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.11}
\begin{document}
\begin{tikzpicture}
\begin{axis}[domain=0:1]
\addplot+[red] {x};
\addplot {x^2};
\addplot {x^3};
\addplot {x^4};
\end{axis}
\end{tikzpicture}
\end{document}

So, for the desired plot style (red with filled circular marks of the same color), you can use either approach:
\documentclass{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.11}
\begin{document}
\begin{tikzpicture}
\begin{axis}[domain=0:1]
\addplot[red,mark=*] {x}; % red and filled circular marker, OR
\addplot+[red,mark=*,mark options={fill=red}] {x^2}; % override cycle list style
\addplot {x^3};
\addplot {x^4};
\end{axis}
\end{tikzpicture}
\end{document}

As to your request in a comment, the complete lists of markers and linestyles available in pgfplots are in Section 4.7 of the manual. Section 4.7.1 lists the markers and available options for styling them, while Section 4.7.2 does the same for line styles.
Rather than specifying the style for every individual plot command, it is also possible to use user-defined or pre-defined cycle lists to set the style for each successive plot command. The default cycle list was shown in the first sample. Here is one other pre-defined cycle list, exotic:
\documentclass{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.11}
\begin{document}
\begin{tikzpicture}
\begin{axis}[domain=0:1,cycle list name=exotic]
\addplot {x};
\addplot {x^2};
\addplot {x^3};
\addplot {x^4};
\end{axis}
\end{tikzpicture}
\end{document}

Other choices for pre-defined cycle lists are documented in Section 4.7.7 of the manual, and the second half of that section describes the syntax for defining your own cycle lists. A user-defined cycle list could be defined once per document and used for all plots in that document. This way, if you change your mind about the style, you only need to update the definition once and all plots will update accordingly.
[red,mark=*]. – Paul Gessler Jul 05 '14 at 18:31cycle lists described (with examples) in Section 4.7.7. – Paul Gessler Jul 05 '14 at 18:52