Yes, the nature of a host's OS plays a big part in how it can be attacked.
Here's the quick 101, but this is a big enough area that the best I can do it point you on the road to further research...
Every OS will have some similarities in architecture - after all, they all server the same purpose of providing a useful abstraction for applications that run above it and a management capability for resources that have to be optimized - like access to memory, CPU utilization, various I/O streams, etc. So there's always some similarites, but even at a low level, OSes have different approaches to even some very basic functions.
The very nature of how the basic features of an operating system are put together forms the baseline for what attacks are likely to be the most successful. In any OS, the star of the show, from a security perspective, is the Trusted Computing Base (TCB). This is the combination of hardware and software on which the security of the system depends. This is classically definted as the kernal and its most trusted processes.
In almost any operation, you'll see that an application has to use the TCB to do something of importance, so a big part of OS managemenet is allowing proceses to make use of TCB capabilities, moving from a state of low trust to high trust. How that's negotiated and the basic premises involved in this element of the architecture are central to understanding how to both protect and attack a given OS.
A really, really interesting corrallary to the vulnerability of an OS is it's market share. I can never really believe an OS manufacturer when they say "well... we have less vulnerabilities than X" - because the number of known vulnerabilities is not something is derived from an academic lab (most of the time) - a great number of exploits come from the wild - and the most well tested (and exploited) operating systems are the ones with the greatest market share.
Attackers won't go after an archaic OS - there's no gold or glory in a small user base or a minor deployment with nothing of value. As an OS rises in popularity, it's not surprising that hacks of the OS tend to rise as well.
You can spend months really digging into the nature of the Trusted Computing Base and it's implementation in the various popular OSes. To do so, the best avenue is to dig into a few books on "Computer Security", or "Operating System Security" - there's quite a few books out there that will take you through a fairly abstract path to understanding the basic premise of a TCB and the fundamentals of it's implementation in a few major operating systems. You may or may not wish to fall back to a fundamental book on OS basics or grab a college course on this. It's an area that doesn't change in the basics so very much that a college professor would be monumentally out of date.
Then, if you're still tantalized, I recommend digging into the manufacturer's documentation. In many cases, bigger OS manufacturers have had to go through various types of security certifications to get the bigger government contracts. That usually includes publicly available documentation, such as what's available with the Common Criteria. This link provides all successfully evaluated products, and there's a pull down that will let you choose "Operating Systems" to get just those OSes that have passed review. Each one includes links to details about how the evaluators evaluated the product and what they came up with. The goal of a rating like this is to give consumers an independant assessment by an expert of what security protections a given product can and cannot provide. The value of the evaluation is a topic for a whole other debate, but what I like about it is that it's a (somewhat!) consistent review process, so it lets you see the writeup of different products in a single category.
WARNING: These are dry, dry, dry. I usually have to glue my eyelids open, but in all honesty, this is heavy reading and I don't see a way to make it light.
Lastly, I'd just out vulnerability alert systems. You'll see that attacks can be aimed directly at OSes, or at a combination of application and OS. I'm a fan of NIST's simply because it has a search window and I'm relatively familiar with other parts of NIST's work, and I like the source databases it pulls from. A quick run of "LINUX", "Mac OS X" and "Windows" will show you plenty. One thing that's interesting is in some cases, it's a vulnerability in multiple OSes, because it's really targeting an upper layer application (like Adobe Flash) and then exploiting the OS from there.
You can also check vendor websites for the patches themselves. Almost everything you see on the list above will result, sooner or later, in a patch. But it varies wildly how quickly a given patch can be fixed.
It's a fine exercise, but realize that each vendor has their own motivation to give the impression that their security is well under control. So you may not hear the most useful information here. You will hear good advice on how to lock down a given OS, however.