Ubuntu will run inside your existing Windows operating system. You'll be able to simultaneously run Ubuntu while using other Windows applications. However, Ubuntu will run quite slowly, in its own box (which albeit can still be maximised) and it won't be representative of an actual Ubuntu installation - not making full use of your hardware. It will be frustratingly limited, IMHO, if you want a full desktop experience. It may, however, be useful if you want to use Windows as your primary operating system but you need to run one specific Linux program, or a Linux server, for testing purposes.
Ubuntu will run natively on your machine, separately from Windows. You won't be able to run the two simultaneously, but you can choose which one to run when you boot. This is a good option if you want to get a full desktop experience, as long as you are able to resize some existing partitions to make space and you are reasonably confident in doing so (that is, you won't trash your existing partitions, and you are good with backups). If you are thinking of moving to Ubuntu permanently in the future, this is the only option of the three that will let you do that smoothly, since you will be able to remove or shrink your Windows partition later without affecting Ubuntu.
Once installed, this is fairly functionally the same as running Ubuntu in its own partition: you won't be able to run Ubuntu and Windows simultaneously, Ubuntu will have full access to your hardware and you get to choose Ubuntu or Windows when you boot. This option, however, is for when you don't want the permanence of re-sizing any existing partitions. It is good for if you want to try Ubuntu temporarily without changing any partitions. The trick here is that special magic allows you to create the Ubuntu partition as a virtual partition inside your existing Windows partition, rather than creating a real, on-disk partition. Ubuntu will see it as its own partition though, and Windows will see it as a normal file. You won't be able to delete your Windows partition though, which is why I said it's good if your use of Ubuntu is temporary. It is plausible that disk access might be slightly slower using this method than having Ubuntu in its own partition, but this may not be noticeable and in all other ways Ubuntu will run just as fast.
Also, if I choose to partition, should I partition the hard drive myself or should I let the Ubuntu installation menu do it for me?
This in itself is a rather broad question and ten different people will have ten different opinions on this. In my opinion you may as well let Ubuntu choose its own partitioning. This will result in small partition for swap and a larger one for everything else. Some say it's worth having a separate /home partition. You can do this if you like, though if you don't know you need it, though, you probably don't really need it. The benefits of a separate /home to some people is that they can trash everything but keep the /home partition, so their files remain, when they want to do a clean reinstall. Of course, if you have backups of your files, then you could choose to restore from backups instead.