Peak detection has quite a few applications, for 1D or multidimensional signals. Here are a few examples showing how varied these signals and their interpretations of a peak can be:
The original poster's 1D data;
Hough transform of an image, each peak corresponds to a line in the original image;
autocorrelation of an image, each peak corresponds to a frequency revealing a "periodic pattern";
"generalized" cross-correlation of an image and a template, each peak corresponds to an occurrence of the template in the image (we may be interested in detecting only the best peak or several peaks);
- result of filtering an image for Harris corners, each peak corresponds to a corner in the original image.
These are definitions and detection techniques of peaks I have encountered--certainly there are others that I either forgot or don't know, and hopefully other answers will cover them.
Preprocessing techniques includes smoothing and denoising. @Mohammad's answer is about wavelets, and you can see various usages of them in the documentation of Mathematica's WaveletThreshold (where I also took my examples from, by the way).
Then you search for maxima. Depending on your application, you need only the global maxima (e.g., image registration), a few local maxima (e.g. line detection), or many local maxima (keypoints detection):
This can done iteratively, looking for the highest value in the data then erasing a region around the selected peak, etc. until the highest remaining value is below a threshold.
Alternatively, you can look for the local maxima within a certain neighborhood size, and keep only those local maxima whose values are above a threshold -- some recommend to keep the local maxima based on their distance to the rest of the local maxima (the further the better).
The arsenal also features morphological operations: Extended maxima and top-hat transform can both be suitable.
See the results of three of these techniques on an image filtered for Harris corners:
Moreover, some applications attempt to find peaks at sub-pixel resolution. Interpolation, which can be application-specific, comes handy.
As far as I know, there is no silver bullet, and the data will tell which techniques work best.
It will be really nice to have more answers, esp. coming from other disciplines.