As much as I'd like to say yes, I don't think so. They are tuned to our tone of voice, facial expressions, and body language, but apart from familiar sounds like "puppy dog treat" and "you wanna go out", they don't even know what words are. They just associate familiar sounds, like their name, with certain reactions from us. It's training.
I know Warren is correct about pets acquiring a vocabulary...at least my dogs have quite a large one, and it's not just the tone of my voice. The tone can change their mood, as in getting them excited or subdued, but these guys know a lot of my words. For instance, they know when I'm getting out of bed to go to the bathroom as opposed to getting up for real. Three of them sleep with me, and if I have to go to the bathroom and will return shortly, I say to them in a normal speaking voice, "I'll be right back. I'm just going to the bathroom." They stay in bed. If I don't say that, they assume that I'm getting up for real, and they jump out of bed with me. I really believe that they comprehend many words and even phrases.
Oh, and I understand much of their language as well.
YES and you have to be careful what and how you say things! Two examples:
I have an old cat, Pie Face. Two words she understands and uses are "MORE" and "NOW". Of course both refere to FOOD! I realized one day that I was using the wrong word to tell her to wait. When she said "MORE", I would reply "No not now!". To which she would answer more excitedly, "NOW!" I changed my phrase to "Soon!" She lays her ears back at this 'Hissy' word but she knows she will not be getting food.
Over the years, the cats have taught me a few 'Catlish' words that have been passed down to other cats. One of my cats, Chuck would sell her tail for Turkey or Ma HE in Catlish. One day, I got a frozen Turkey Breast out to thaw and Chuck saw me. I said to her, "We will have to cook the Ma HE later." and went on about my business. A couple of hours later everyone (cats) came into the kitchen wanting food so I doled it out to all. I noticed Chuck was NOT eating so I ask her, "Chuck, what's wrong?" She looked at me and answered, "Ma HE!" I realized she was waiting for the turkey! I explained we had to cook it first (a concept she knows but does not quite understand) and hearing that she ate the food I gave her.
I don't think my dog understand exactly what I'm saying to him, but I do think he likes the fact that I'm talking to him! I think he understand some things... I know he knows what the word "cat" means =) He understand tone I think more than words. He knows when I take a stern tone with him that he's done something he shouldn'tve and when I speak in a happy tone that everything is good!
Do they understand communication, sure. Do they know what we're saying, not quite. They associate words (sounds) with certain things and behaviors. They're great at reading body language (including yelling and baby talk) but no, I don't believe they understand English quite the way many of us would like to believe.