Welcome to the club Sally. I bought my iPad a couple years ago after a long series of windows-based computers and smart phones. It took about a week or so before I got the complete hang of it. As much as I hate Apple.inc, their devices are extremely user friendly once you get the basic concept.
Now all the other guys have done a great job answering but I'll throw in my two cent worth
the little black square that appears on the desktop is the icon that indicates a new app is in order to be downloaded (if I got you right, you were talking about a little black square that shows up right after the already installed apps). However, I'd have to agree with the guys that since you didn't get a box asking for a password then you can't be downloading. I'm inclined to think the iPad is set up under the previous owner iTunes account and thus you can access the app store but I don't think you are actually communicating properly (probably the previous owner kept his account but unauthorized this particular iPad).
Here is my suggestion for something you can do till you receive your new cable:
Go to Settings >>>>
Scroll down on the left hand and click on Store >>>>
On the right hand it will show you some options (or not) but at the bottom of the right hand area you will see Apple ID and a name or blank:
- if there's a name then click it, a new window will open, click sign out then sign in by creating a new apple ID
- If no name then just sign in by creating a new Apple ID
The process of making a new apple ID should take like 3 minutes but make sure you remember it, this will be your ID for any transaction or communication with Apple and subsidiaries.
Alternatively, you can create an iTunes account from your computer (with OUT connecting the iPad to it). Check out: support.apple.com/kb/ht2731 for instructions.
NOTE (please anyone correct me if I'm mistaken), if you set up an apple account from your iPad you will be required to input a credit card number but if from a laptop you will have an option to bypass this (at least that was when I helped a friend set up his account a couple years ago).
After creating your iTunes account you can use the steps I mentioned above to log into the store.
After that, just start up the app market and download to your heart's content
Now with this long post it may look complicated but just last night I helped my sister set up her new iPad and it took around 10 mins for the whole deal.
Hope this helps... Cheers