If you have to - or if you want to - use a lightweight web browser you have to be ready to live without support for Java, Flash, and even cascading style sheets. Your old computer did not support Java or Flash a decade ago, and you just cannot expect to have all the modernities if you are using a very old computer, let us say with less than 64 MB RAM. If you have more memory, consider using for example Opera.
There are several lightweight and minimalistic browsers available. Some of them can be used only in text mode.
Lynx is one of them. Lynx was originally written for Unix and VMS, It is available for many platforms, including DOS, BeOS, MINIX, QNX, AmigaOS and OS/2. I use it daily when following links in the emails I receive, as I have configured my email client pine to use lynx as web browser.
w3m is another text mode browser. It has English and Japanese help files and it can be used in both languages. w3m can display tables and frames. In addition, it can be used in pipes as a pager, just like more or less are used.
Links is a text mode WWW browser with ncurses interface. It supports colors, tables and background dowloads. Frames are supported as well. It is configured by menus.
Links can be used not only in text mode, but also in graphical mode by starting it with option -g. It might be good enough for simple graphical pages.
Elinks is based on links. It is a text mode WWW browser, supporting colors, table rendering, background downloading, menu driven configuration interface and tabbed browsing.
The official home page for elinks has more screenshots!
Dillo. I suppose Dillo is the lightest graphical web browser. It is pretty simple and unfortunately its development has been frozen. Enclosed screenshot is borrowed from Dillo's home page, as Dillo has already been left out from openSUSE's repositories.