I've visited each of them personally, so my opinion is informed yet subjective. Enjoy!
Aida - This is the Viennese coffeehouse on the cheap. The interior is pink, the feel is kitschy, like out of a fake version of the 1950s (think the set of Grease) but the cakes are delicious, and the quality is every bit as good as a more expensive joint.
Slice of Cake (Biedermeyertorte) €2.90
Herrengasse/Strauchgasse, 1010 Vienna
Slice of cake (Apfelstrudel with Vanillesauce): €6.50
Café Diglas - I went here right before Christmas after seeing Die Zauberflöte at the Staatsoper with my friend A. who is Austrian, but had never seen an opera before - there's a first time for everything, I guess! We ended up getting Verlängerten, which came with homemade house-shaped sugar cookies - nicer than the typical Speculaas you get in a plastic wrapper - though this place is right downtown, so it's pretty pricey. As pricey as Sacher, etc. Nice for a treat. The interior is also magnificent.
Wollzeile 10, 1010 Vienna
Café Dommayer - this cafe is on the "outskirts" of town, i.e. not smack dab in the 1st District (it's in Hietzing, in the 13th). I was invited here with one of the teachers I work with, and the atmosphere is nice, with plenty of "regulars." The cakes are supplied by a chain, Oberlaa, so unlike other cafes, the cakes are not made on-site.
Dommayergasse 1, 1130 Vienna
Slice of cake (Zitronentorte): €3.40
Demel - the most wonderful cake place in Vienna: they always have spectacular displays out:
Kohlmarkt 14, 1010 Vienna
Café Hawelka - This cafe is delightful, very old-style with a cozy interior. They have simply a chalk board with the daily specials - nothing a la carte, like other Vienna cafes. The founder, Leopold Hawelka, recently retired at age 100 (if you know German, you can read his bio here), and personally greeted guests while he was still running the cafe. Because of Leopold's passion for art, the cafe retains a Bohemian vibe.
Dorotheergasse 6, 1010 Vienna
Slice of cake (Marillentorte): €3.90
Kleines Café - this cafe is named for the short, narrow building in which it's located. Very homey and inviting, the cafe is actually much larger than it looks from the outside, mainly due to a recent addition of a back room. The best place to sit is up front, though - right near the bar, or the door. Sitting down for a cup of coffee here, I feel like I've stepped back five hundred years to the middle ages. (I don't care whether a cup of coffee and a piece of cake would be an anachronism - a glass of wine, then!) Whenever I want to get here, I always get lost, and whenever I'm not looking for it, there it is. Sort of like the Room of Requirement in Harry Potter.
Franziskanerplatz 3, 1010 Vienna
Slice of cake (Linzertorte): €3.90
1/8l Grüner Veltliner: €3.10
Café Landtmann - One of the most famous in Vienna, this cafe is right across from the Burgtheater. Funny story attached to this one: my friend and I were trying to find an Easter market (like the famous Christmas markets in Vienna, just not as famous). We asked an older guy who looked like a police officer (but was really just a security guard) if he knew where they were. He had no idea, but asked us if we'd like to go for a coffee with him, since he was just getting off work. We said sure, and he led us to Cafe Landtmann. I thought he was just a nice older guy - about the same age as my dad - and he had kids our age. Not exactly: turns out he was interested in me, and asked me on a date for the following day. I let him down gently, though my friend and I both appreciated his taking us out to coffee and thanked him kindly.
The cafe gets a lot of traffic from the Burgtheater.
Dr. Karl-Luegner-Ring 4, 1010 Vienna
Slice of Cake (Esterhazytorte): €6.00
Café Mozart - this cafe is kitty corner from the Albertina Museum and the opera: the perfect place to unwind after a performance or museum trip. Like the Landtmann, it gets a lot of theater traffic because of its location. The cafe is quite adorable, and the waiters are friendly...relatively speaking, that is. Also because of its location (the Albertina and the opera are two of my favorite places in Vienna), I've spent plenty of time here. Their Apfelstrudel is superior to the one at Cafe Central, though I would really recommend the Kardinalschnitt, sort of like an amaretto-cherry tiramisu - yummy!
Albertinaplatz 2, 1010 Vienna
Oberlaa - this is another chain of bakeries (of which Café Dommayer is one). I've had their macaroons (LAAKronen, as they're called) at the Naschmarkt location, and they were tasty, but nothing to write home about, though they do have nine varieties. I recommend skipping the Naschmarkt (too crowded - I got the cookies to take away) and going to one of the other locations, for a leisurely cup of coffee the "right" way - that is, the Vienna way.
Various locations: http://www.oberlaa-wien.at/de/standorte.html
LAAKrononen: €6.30 for 7 pieces (€0.90 each)
Café Sacher - part of the famous Sacher Hotel, and home to the "original" Sachertorte. This was my first cafe experience in Vienna when I went in 2007. I've been back several times, mostly for the fame effect, and I've made the Sachertorte myself at home. It always seems to turn out dry - not my fave. Word to the wise: it turns out dry in the cafe, too, so do yourself a favor and order it with a whopping side of whipped cream - or get a Kardinalschnitt instead. The famous cake is nothing special, in my humble opinion.
Philharmonikerstrasse 4, 1010 Vienna
Slice of cake (Sachertorte): €5.50
Café Sperl - This cafe is within walking distance from my apartment, and mostly for that reason has become my favorite. A weird thing to note: according to legend, it was also Hitler's favorite cafe in Vienna - yikes! I'm hoping that's all the Führer and I have in common. Another funny story, with a friend: we went here for brunch one unassuming Thursday morning, and this Dutch guy comes up to us and asks to take our picture because the lighting is "just so good!" He's doing a PR campaign for the cafe, and tells my friend she looks just like a Vermeer painting. What a compliment! One more thing: the Sperltorte, a chocolate-cinnamon cream cake, beats the pants off the Sachertorte.
Gumpendorfer Strasse 11, 1060 Vienna
Slice of cake (Sperltorte): €3.50
By the way, most cafes in Vienna do have a full bar, lunch and dinner specials, as well as cakes and coffee. I rarely eat out in Vienna, to save money but also because I like to cook, but - that should not discourage others! I just wanted to mention that most places mentioned do "real" food in addition to desserts. The exceptions would be Aida, Oberlaa and Demel, becuase they specialize in cakes and confections.