Solutions without flash

1. Ajax upload

Very nice tool, flash-free. You can customize it as you want, also you can use drag&drop files on input
Works in: IE6/7, FF2/3, Safari, Chrome, Opera.
Has different versions for jQuery, Prototype, plus has independent version.
demo

2. jQuery Multiple File Upload Plugin

Has many features, like using different validations, limit max files for upload, use different icons for different file types
demo

3. Ext.ux.FileTreePanel (Ext.JS extension)

demo

4. noswfupload

Multiple files upload without SWF objects, applets Java, or other third parts plug-ins, 100% browser and W3 compatibility thanks to graceful enhancement, no extra libraries dependency

demo

Solutions using flash

1. SWF Upload

Popular solution, allows to customize all upload elements using XHTML & CSS, shows only file types you choose, has js callback function to every event.
demo

2. uploadify

Has not so many features, as SWFUpload, but it uses jQuery :)
demo

3. FancyUpload

FancyUpload - fully customizable uploader with progress bar. Require MooTools.
demo

4. Yahoo File Upload

demo