The major problem for German intelligence were ferocious inter-factional rivalries within the military and particularly between the Wehrmacht and SS that prevented effective co-operation and co-ordination.

This goes to /u/madbunnyrabbit's follow-up question below: no, the Germans didn't have a direct counterpart to the GC&CS at Bletchley Park. Instead, they had an assortment of departments within larger commands: the foreign ministry, the RSHA (the Reich Main Security Office within the SS), the OKW (Oberkommando der Wehrmacht, the military high command) and each of the individual Wehrmacht services all had their own cryptographic and cryptanalytical units. This site has good summaries of each of the main German cryptographic units.

In fact, the fragmentation of German signals intelligence is a symptom of a wider malaise in German intelligence during this era: internecine rivalries and a lack of central coordination leading to duplication of responsibilities and ultimately severely damaging the Nazi state's ability to effectively gather intelligence. The vicious, longstanding rivalry in foreign human intelligence between the military intelligence service Abwehr and the SS' Ausland-SD, and both services' lacklustre wartime records offer another example of these manifest problems.

German codebreakers did achieve some notable successes in breaking Allied codes — the cracking of British maritime ciphers by B-Dienst, the Kriegsmarine's cryptography department, was of particular importance to the successful invasion of Norway in 1940, and to U-boat operations during the Battle of the Atlantic.

Soviet SIGINT and communications security were seriously underserved during this period, and the field SIGINT units of OKH (Oberkommando des Heeres, army high command) succeeded in penetrating a large volume of tactical traffic during Operation Barbarossa and the subsequent Eastern Front campaigns. Richard Aldrich's superb history of GCHQ has an interesting section on the Anglo-American Target Intelligence Committee (TICOM) teams that were tasked with capturing as much of the German SIGINT apparatus — and, crucially, their knowledge of Soviet ciphers — as possible in the final months and immediate aftermath of the war.

But overall, German successes paled in comparison to that of the British and American effort. It wasn't that Allied codes were 'better', per se; rather, what the Germans notably failed to achieve was any kind of consistent, high-level penetration of enemy military and diplomatic traffic. Their successes — as with Bletchley Park's breaking of Enigma — were usually enabled at least in part by capturing codebooks and cryptographic equipment on the battlefield. And just as at Bletchley Park, these intercepts could quickly dry up if the enemy became aware that their communications were compromised. The cryptographic advantage was never permanent; part of the reason we remember Ultra as a great success is the caution that Allied commanders exercised in acting on Enigma intercepts.

The Germans experienced this first-hand during the North African campaign: in 1941, OKW/Chi broke a US diplomatic cipher ('Black'), enabling them to read cables sent to Washington by the American military attaché in Cairo. This gave Rommel a major tactical advantage — but then two disasters struck. First, the Afrika Korps' SIGINT battalion was effectively wiped out during the Battle of Gazala in July 1942; then, the British learned from a PoW that their communications were being intercepted. Both the military attaché in Cairo and his cipher were replaced, and the near-real time supply of tactical intelligence on Allied operations in North Africa ceased. That, coupled with an increase in during mid-1942 the quantity of Ultra intelligence received from GC&CS handed the British a strategic advantage that helped them win the Second Battle of El Alamein.

I recommend David Kahn's The Codebreakers for an excellent read on the history of cryptography and cryptanalysis — it includes a couple of good chapters on Axis SIGINT operations during the war.

Edit: /u/thefourthmaninaboat's comment from the FAQs, which I hadn't seen until just now, does a rather better job than I did here of explaining what ciphers the Allies actually did use.

I'm phone-posting, so can't easily find the link, but I've written an answer to this question, which is in the FAQ. A quick recap would be that the British Army and RAF used the TYPEX machine, a cipher machine based on Enigma, but with improvements that made it far more secure. The RN started off using book codes - substitution ciphers with instructions given in code books. They switched to TYPEX mid-way through the war. The Germans never broke TYPEX, but the German Navy's SIGINT group, the B-Dienst, were able to break most of the RN's codes. The Americans used a very sophisticated cipher machine called SIGABA, which was all but impossible to break at the time. An attachment to SIGABA and TYPEX was produced, called the Combined Cipher Machine. It allowed TYPEX messages to be decrypted on a SIGABA machine and vice versa.

You might like this previous question and its answers

As a follow up question.... Did the Nazis have any sort of equivalent to Bletchley park or did they just assume Allied codes were unbreakable?