A Beginner's Guide to Beethoven's Compositions

His most prized compositions include:

     Nine symphonies. 

Nos. 3, 5, 7, and 9 are the most widely praised. Symphony #6, Beethoven's symphony about nature, also has many admirers. The other four are very good, too, although No. 1 is the least "Beethoven-ish." (Learn more about the symphonies here)

     Concertos.  His third, fourth and fifth piano concertos are highly regarded as is his one violin concerto.  His Triple Concerto" for violin, cello and piano is also noteworthy.  (listen to the concertos here)

     String Quartets.  Of the 16 quartets, the five "late" quartets (1825 - 1826) are considered the most profound ever written, but they are usually difficult for beginners to appreciate.   Easier are:  the six Opus 18 quartets (1801), the three "Rasumovsky" quartets (1806) and the "Harp" (1809) and "Serioso" quartets. (listen to the string Quartets here) 

     Piano sonatas.  Like the "late" String Quartets, Beethoven's five "late" sonatas, Nos. 28 - 32 (1816 - 1822), are considered the pinnacle of sonata composing.  Again, the earlier sonatas may be more approachable for the beginner.  One way to tackle them is to listened to the "named" ones: the "Grand" (1796), the "Pathétique" (1799), the "Moonlight" (1801), the "Pastoral" (1801), the "Tempest" and the "Hunt" (both 1802), the "Waldstein" (1803),the "Appassionata" (1805), "à Thérèse" (1809),  "Les Adieux/Das Lebewohl" (1811).  Or just listen to any sonata between numbers 7 and 27 except for 19 & 

20 ("early" sonatas that were published later).  (listen to the piano sonatas here).

     Miscellaneous.  Beethoven's last three violin sonatas are very good.  (listen to the sonatas here).  His second mass, Missa solemnis (1824), is one of the greatest masses ever written (listen here), and while it is debated whether his one opera, Fidelio (1804 - 1814), is a "great" opera, it is at least highly respected (watch here).

His bagatelles (short piano pieces written to be played at home by amateurs) are very popular, especially his very famous unpublished one, Für Elise.  Example bagatelles:  Opus 33), Opus 119 , and Opus 126).  

Beethoven also wrote several cycles of German lieder (songs for one singer and piano).  You can listen to An die Ferne Geliebte, op.98 with English subtitles here. 
 
Finally, besides the string quartets, you can explore some of his other chamber works here.