Tel Aviv is called "The city that never stops" by tourists and locals alike. It has a wide range of Pubs, Bars, Clubs and it is known worldwide for the Pub Scene of Tel Aviv. The entire city is crawling with nightlife attractions and you would actually have to work pretty hard to find yourself further than 500 meters away from a place to have a drink. People from all the surrounding region come to Tel Aviv to have a drink or a party so on weekends traffic is hectic at late hours and finding a parking spot is somewhere between hard and impossible (so sticking to cabs is not a bad idea). Any day is a good day to party in Tel Aviv, not just the weekends.
New places are opening and closing every day and the "hottest spots" change every couple of months, so no internet guide will be able to direct you to the hippest place (even though some may try). Many places in Tel Aviv have minimum age limitations that vary from 18+ to 30+. Usually the limitation is different between males and females and while some spots may be flexible others will be as strict as possible.
Israel has no unique drinking culture so any place with any self-respect will have the entire world wide alcohol selection available, from Wine and Beer to Tequila, Arak, Vodka, Whiskey and Cognac. One of the most popular drinks is the local Goldstar beer and the Annis based Arak.
Even though the entire city is full of spots to hang out, there are a few places that have an unusual amount of pubs/clubs:

Tel Aviv seaport - Located at beach side to the west of the Yarkon Park right between Tel Aviv center and north is the old seaport. The entire place is full of clubs, pubs and restaurants right next to each other door by door. Notable places: TLV Club, UpTown, Erlich, Shalvata, Seabreeze, Whiskey a gogo and more. Very busy in weekends during the summer and on warm days during the rest of the year, as this area attracts people from all around the city and the wider Gush Dan area.

Dizengoff - Ben Yehuda St. - The north ends of these streets are full of chic bars that are packed almost every day with a 22+ crowd. Sometimes it's just hard to breath there. Notable: Friends, Bergman, Rosa, Yermiyahu.and "223" bar for more of the new york speak easy cool spoot

The Boardwalk - The entire beach area from the seaport in the north to Jaffa in the south is full of mainly cafes, restaurants and bars. Some are normal open bars while others actually spread to the beach with tables on the sand. This is the more "touristy" area of Tel Aviv's nightlife scene, that the "real Tel-Avivians" try to avoid.

Allenby St. - Going from the Beach to the west all the way to the south-east of Tel Aviv, Allenby is one of the longest streets in the city. The western area is full of mainly pubs and dance-bars, not the hippest clubs but stable places that have been there for years and are occasionally full of tourists. Allenby Street may sometimes feel a bit dodgy but fear not. It's cheap but mostly not recommended to eat.

Lilinblum - Levontine - Nahlat Binyamin st. - A few streets around the east side of Allenby with many trendy pubs with an extremely sophisticated crowd, and many dance bars that range from the bluntly commercial to the leftfield indie. Any arrivals to this area will ensure a good drink. Notable: Shesek, Lima Lima, Atara, Betty Ford, Bordel, Flame, Academia, Abraxas, Minus one and more.
Ha'Masger - Ha'Rakevet St.' - Mainly a clubbing area for Tel Aviv's younger crowd (18-19) with huge clubs and dance bars. Notable: Dome, Vox and more.

Florentin - Mostly small neighborhood bars for a cool fun night out in a chic area in Tel Aviv. Most spots in Florentin appeal to the artsy and indie crowd. Florentin has a "rugged" appearance, especially at night, but it is totally safe. Notable: Hudna (Abarbanel street), Comfort 13, Haoman 17 and all the little places on Florentine st. and Vital st.

King George-Tshernechovsky - in the close to Shenkin st. upper side of King George you can find some alternative cafes and bars, like "Geatzel Shapira" on Almonit lane and "little prince" which is the center of the young poetry revival movement that connected to "Maayan" poetry magazine and others interesting poetry or art fanzines. On Tshernechovsky, not far from there, there are several cafes and cheap restaurants. close to Dizengoff Center, you can find "Bacho" cafe, a nice place with too-artistic atmohphere, "Hakosem Falafel" and the "Yemen Falafel", both recommended.

Even Gvirol - A lately very developed pubbing area with some of the coolest pubs in Tel Aviv. During the day appeals to the many lawyers and businessmen working in the area. Notable: 2 clubs - Vila Sokolov and Landen, and the pubs-restaurants Dorothy Gale, Brasserie and Liliroz.

Habarzel (Ramat Ha'Chayal) - Located at the north near the rich neighborhoods. This area has been developed to accommodate the vast high-tech industry around it, so one can expect somewhat commercialized and rather upscale spots. Notable: Leo Blooms, Molly's, Frame, Sushi Samba, Giraf, Black, Segev Express, Moses, Max Brener.

Karlibach - A new clubbing area with pubs growing in every corner. Notable: ZiziTripo, Hachatul Ve'Hakelev.

Famous Tourists Spots

Mike's Place, 86 Herbert Samuel (Next to American Embassy). An American style bar located right by the American embassy that features live music every night of the week. Also features outdoor seating in the more pleasant weather, pool table and televised sporting events. Mostly Anglo 20-30 something crowd, very good bar with several kinds of beer on tap.

Molly Bloom's Irish pub, 2 Mendele St.. The first Irish pub in Tel Aviv. The pub has a great atmosphere and reasonable prices, and is quite busy on weekends. Also, it's close to the hotels. Usually hosts many people from the UK and from the Republic of Ireland.

The English Bar, (Allenby st. near the beach). A UK based sports bar and if you happen to end up there during a Premiership game, you're in for a native UK experience.

Mendalimos, Hayarkon 102 (infront of the Dan Hotel), ? +972-508-464462 (yairneumann11@gmail.com). An Israeli pub and dance-bar. This is where a lot of the locals go to just to grab a beer or to enjoy the nightlife scene. Vintage design indoors, alongside outdoor balcony in which all the most important sports games are broadcast. Trendiest music from abroad and Israel.