The most popular.oldest directory is dmoz and they manually approve submissions so it can literally take years or you may never be approved. Others as neil mentioned can differ in time frames, personally though I think investing in social networking, blog commenting etc...are much more effective these days than directories. Directories can even be harmful to your site if google starts calling them "link farms".
create an xml sitemap and leave it at www . yourdomain . com / sitemap.xml and submit that sitemap to search engines. DMoz is for PR3 + only so if your site is PR3 + it will be listed free, if below 3 then there is no light at this end of DMoz.
Every directory have their own rules, just read them and you will know how much time they need to do review.
Most high page ranked directories need much more then month to add you, some of them need even 6 months.
If you pay your site will be listed in 24-48h.
If you add your website to DMOZ (and follow all "how to add website" rules) they will review your listing form 1week to several months (there is no rules for this).
Really depends chap, all i will say is don't take up the offer of recipricol linking in exchange for a listing in 1to2 weeks. either pay for the listing (not expensive normally) or do the free link which usually takes a while
Scope of doing Directory submission:
-Help in getting Good quality of back links to our website which leads to increase in getting traffic to our website.
-Help in increasing the Page Rank of our website.
Directory submission basically listing your website in various business directories that best suited your services. Directory submission was widely practiced by SEOs because it was very effective to get a good page rank and good position in the SERPs.