You have Cat icat (should be iCat, but that is ok) listed in your instance data. You then have your remove method that also declares a local instance of Cat called iCat. iCat has to be something. I know that iCat is declared as an object from another class called Cat. You already have an instance of Cat in your array of Cat right? So just use that in your remove method:
public void removeCat( String iname )
{
// need to also use the .equals method here because you are
// not doing a boolean comparison you are comparing two
// string objects
///** if( iname == icat.getName() ) **
for (int i = 0; i < aKennel.length; i++) {
if(iname.equals(icat[i]))
{
myCatNumber--;
System.out.println( iname + " has been removed." );
/*also you will have to do some serious array manipulation here to resize your array and actually remove the cat object that the user chose. I suggest that if you are just trying to get this to work use an ArrayList rather than an array If you want to do it this way you will have to either use the copyArray method or create a "temp" array that does the actual remove and then iterate throught the array again and copy the left over values of the temp array into the aKennel array*/
}
}
System.out.println( "Sorry, we don't have any cats by that name." );
}
}
Plug away and try it out. It will take some time and some practice, but you have the right idea.
A good resource that has been there in the past for me when I was learning java is www.javaranch.com They have beginners and intermediates forums that you can post to and they are always more than eager to assist.
Hope this helps you out.