This is a potentially huge question with almost limitless answers. I enjoy the artistry of many conductors in a wide variety of repertoire and for many different reasons, as one might guess. Ill cover only a few.
The Kleibers are both favorites of mine. Erich Kleiber is very much a product of his age and his recordings manage to combine old-world breadth and depth with more contemporary drive and focus. Hes at his best in German standard rep LvB, Mozart, Strauss, etc and then in opera, too a Le Nozze di Figaro, Der Freischutz, Der Rosenkavalier. His son Carlos is more modern, generally tends to faster tempi, and generates at times unparalleled excitement. His very best concert music recordings LvBs 4th & 7th (Philips DVD), 6th (Orfeo), Brahms 2nd (Philips DVD) are among the finest committed to disc. But his best work is in opera. His 1979 Munich production of Der Rosenkavalier (DG DVD) must certainly be one of the greatest recordings ever of anything (and his other recordings of the work, pirated or legit, are all superb, too), his Der Freischutz is comparable to his fathers, and his Tristan und Isolde whether in his DG studio recording or the pirated 1978 La Scala version, is excellent, too.
Another favorite is the underrated Rafael Kubelik. Excellent or better in everything he recorded (at least that Ive heard), his almost always lyrical, always supremely crafted readings of all manner of repertoire are always a joy to hear. His Dvorak is about as good as it can possibly get. His Wagner is superb, Die Meistersinger, in particular. Then theres his Smetana, Weber (including wonderful versions of both Der Freischutz and Oberon), Schoenberg, Martinu, Janacek, Mozart, Beethoven, and of course Mahler. Hes a great all-rounder.
Karl Bohm is an old world conductor of the highest order. His conducting style is no longer in vogue, being somewhat heavier than is now favored, but when one listens to his core German rep Beethoven, Mozart, Schubert, Bruckner, Wagner, and Strauss, especially one hears unsurpassed music making.
For those who like often dramatic (in Mahler especially) but always at least vibrant music making, Leonard Bernstein is one way to go, and for almost inhumanly well crafted conducting, Herbert von Karajan is of course excellent. I tend to prefer his work from the 40s and 50s and perhaps early 60s, and hes still controversial today for his interpretive style a manicured wall of sound but no one can deny his mastery at getting exactly the sound he wanted.
Sergiu Celibidache is an acquired and unusual taste. He mostly scorned the recording studio, so almost all available recordings of his art are taken from concerts. His style can best be described as prolonged and analytical. His earlier recordings are more standard in conception, but in the 80s and 90s he began stretching out his interpretations, focusing on slow tempi in the extreme. Some people cant stand it, but if in the right mood, I love it. In certain works. His Bruckner is sublime, and his Tchaikovsky is very good. He seems to have had a special affinity for Brahms, and both his Stuttgart and Munich recordings of the 2nd rate among my favorites. Despite professing a dislike for vocal music, his choral recordings are very good. His EMI Brahms Requiem is amazing, and his Faure Requiem and Bruckner F Minor Mass are both compelling. But surely it is his scaled down, transcendental Bach B Minor Mass that rates among his greatest achievements. Very slow at times, but very fast at other times, it is simply amazing. Hes not as compelling in some other music, though. His Beethoven can be too long and slow, likewise his Mozart and Haydn. (Though a pirate recording of him accompanying Murray Perahia in Mozarts 9th Piano Concerto is surprisingly taut, if a bit beefy.)
Carlo Maria Giulini is among the very greatest of all conductors as far as Im concerned, and hes at his best in the big works and, especially, opera. His early 80s Eroica is huge in scale, and slow in tempo, but he builds the tension of the piece up to the point where it becomes the massive, revolutionary work it is. His late Bruckner symphonies (7, 8, 9) are all among the very best, his slow and hypnotically powerful Bach B Minor Mass moving in a unique way. But it is in opera where he looms largest. His Mozart opera recordings are quite possibly the best made. Id like to hear of a better Don Giovanni. And surely his Le Nozze is among the best. Better yet is his Verdi. His Don Carlos sets the standard, and his Rigoletto, Falstaff, and Il Trovatore are uniformly superb. Some may not like the studied, serious, slightly slow approach in all these works, but he pulls them all off magnificently.
Among living conductors, theres a lot to choose from. Claudio Abbado is an amazing talent, bringing great analytical clarity and innately musical flexibility to almost everything he conducts, from Mozart right up to living composers. Pierre Boulez, also a great composer, brings analytical precision and clarity almost no one can match, whether in the studio or concert. He tends to be best from Mahler forward (though I love his Wagner), and in some repertoire (Webern and his own music, for instance), he is unrivalled. Esa-Pekka Salonen offers similar strengths, but marries them to greater energy. Michael Tilson-Thomas is more flexible and variable, and his Mahler is likewise variable, but often very good. Colin Davis is the absolute master of Berlioz his two recordings of Les Troyens are the best available, for instance and is equally at home in Mozart, Britten, and Sibelius. Michael Gielen is a modern music master, and like Boulez is at his best from Mahler forward, as is Peter Eotvos. Kent Nagano is fast becoming a favorite of mine, and in varied repertoire. Hes another modern music specialist, as his Messiaen, Schoenberg, and Prokofiev attest to, but his recent DVD of Wagner Parsifal is quite formidable.
There are others, of course, but this is a nice OTTOMH list.