In this paper, we study the following Kazdan-Warner equation with sign-changing prescribed function $h$ begin{align*} -Delta u=8pileft(frac{he^{u}}{int_{Sigma}he^{u}}-1right) end{align*} on a closed Riemann surface whose area is equal to one. The solutions are the critical points of the functional $J_{8pi}$ which is defined by begin{align*} J_{8pi}(u)=frac{1}{16pi}int_{Sigma}| abla u|^2+int_{Sigma}u-lnleft|int_{Sigma}he^{u}right|,quad uin H^1left(Sigmaright). end{align*} We prove the existence of minimizer of $J_{8pi}$ by assuming begin{equation*} Delta ln h^++8pi-2K>0 end{equation*}at each maximum point of $2ln h^++A$, where $K$ is the Gaussian curvature, $h^+$ is the positive part of $h$ and $A$ is the regular part of the Green function. This generalizes the existence result of Ding, Jost, Li and Wang [Asian J. Math. 1(1997), 230-248] to the sign-changing prescribed function case. We are also interested in the blow-up behavior of a sequence $u_{varepsilon}$ of critical points of $J_{8pi-varepsilon}$ with $int_{Sigma}he^{u_{varepsilon}}=1, limlimits_{varepsilonsearrow 0}J_{8pi-varepsilon}left(u_{varepsilon}right)<infty$ and obtain the following identity during the blow-up process begin{equation*} -varepsilon=frac{16pi}{(8pi-varepsilon)h(p_varepsilon)}left[Delta ln h(p_varepsilon)+8pi-2K(p_varepsilon)right]lambda_{varepsilon}e^{-lambda_{varepsilon}}+Oleft(e^{-lambda_{varepsilon}}right), end{equation*}where $p_varepsilon$ and $lambda_varepsilon$ are the maximum point and maximum value of $u_varepsilon$, respectively. Moreover, $p_{varepsilon}$ converges to the blow-up point which is a critical point of the function $2ln h^{+}+A$.
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