I'm going to spare myself a lot of typing by NOT defining what a conversion is on this site. If you're at a point where you want to buy PPC ads on Google or elsewhere, or if you have a live campaign and just need some help with managing it, then you should know what a conversion is. I will give you this link, as an act of kindness, but that is all.
I'll also define some of the acronyms because, admittedly, they can be a little confusing:
Conversion Options
Google Adwords
Google offers conversion tracking for free to all Adwords advertisers. It's really easy to implement this, and for basic conversion tracking (e.g., leads, sign ups & newsletter subscriptions), it's just silly not to implement it. You can also track conversion value directly in Adwords and this is very useful for e-commerce campaigns where the value of a sale changes for each customer. I recommend you implement Adwords internal tracking even if you're using an outside tracking tool such as Omniture or HitBox because it makes keyword-level optimization a lot easier than if you have to pull third-party tracking information into your Adwords reports.
Google Analytics
In March 2005, Google acquired a company called Urchin which had created a robust Web analytics tool called Urchin Stats. Google rebranded Urchin as Google Analytics and gave it away for free to everyone who wanted it (Urchin charged as much as $300/month for the software). Thus, Google Analytics is a very robust web traffic tracking tool and it's easily integrated with Google Adwords. I recommend implementing Google Analytics if you're not already using something to track your Web traffic. As with most third-party tracking tools, you can also track media campaigns in Google Analytics including campaigns on Yahoo and Bing.
NOTE: The one major downfall to using both Adwords tracking and Google Analytics is that it's difficult to track conversions when they are hosted on a third-party domain, such as a Yahoo Shopping Cart. I'm not a programmer, so it may be entirely possible to do this, but I have yet to figure it out. Google Analytics also offers absolutely no product support beyond the online help files, so if your job depends on reporting your web site and/or campaign results to your boss, I do not recommend using Google Analytics. There are consultants and agencies who specialize in setting up and managing Google Analytics, so that is another option - but make sure you have money in your budget for this.
Third Party Analytics Tools
Many of my clients use third party tools such as Omniture and HitBox to track both their Web site traffic and their campaign traffic. These tools are targeted and mid to large size companies which may have multiple web sites and/or divisions. As such, they're not cheap. There are a number of cheaper third-party alternatives that range from as little as $30.00/month to hundreds of dollars per month. It's worth doing some research if you plan on implementing a new web traffic tool - I personally can't recommend anything other then Google Analytics. I've had a lot of success with it on my own sites and my clients' sites.