The analysis of biological fluctuations provides an excellent route to probe the underlying mechanisms in maintaining internal homeostasis of the body, especially under the challenges of the ever-changing environment or disease processes. However, the features of nonlinearity and nonstationarity in physiological time series limit the reliability of the conventional analysis. Hilbert-Huang transform (HHT), based on nonlinear theory, is an innovative approach to extract the dynamic information at different time scales, in particular, from nonstationary signals. In this paper, HHT is introduced to analyze the alpha waves of human's electroencephalography (EEG), which seemly oscillate regularly between 8 and 12 Hz in healthy subject but getting irregular or disappeared in different demented status. Furthermore, conventional timefrequency analyses are adopted to collate the results from those methods and HHT. Finally, the potential usages of HHT are demonstrated in characterizing the biological signals qualitatively and quantitatively, including stationarity analysis, instantaneous frequency and amplitude modulation or correlation analysis. Such applications on EEG have successively disclosed the differences of alpha rhythms between normal and demented brains and the nonlinear characteristics of the underlying mechanisms. Hopefully, in addition to empower the studies of EEG varied in diseased, aging, and physiological processes, these methods might find other applications in EEG analysis.