Button inside of anchor link works in Firefox but not in Internet Explorer?

You can't have a <button> inside an <a> element. As W3's content model description for the <a> element states:

"there must be no interactive content descendant."

(a <button> is considered interactive content)

To get the effect you're looking for, you can ditch the <a> tags and add a simple event handler to each button which navigates the browser to the desired location, e.g.

<input type="button" value="stackoverflow.com" onClick="javascript:location.href = 'http://stackoverflow.com';" />

Please consider not doing this, however; there's a reason regular links work as they do:

• Users can instantly recognize links and understand that they navigate to other pages
• Search engines can identify them as links and follow them
• Screen readers can identify them as links and advise their users appropriately

You also add a completely unnecessary requirement to have JavaScript enabled just to perform a basic navigation; this is such a fundamental aspect of the web that I would consider such a dependency as unacceptable.

You can style your links, if desired, using a background image or background color, border and other techniques, so that they look like buttons, but under the covers, they should be ordinary links.

Just a note:
W3C has no problem with button inside of link tag, so it is just another MS sub-standard.

Answer:
Use surrogate button, unless you want to go for a full image.

Surrogate button can be put into tag (safer, if you use spans, not divs).

It can be styled to look like button, or anything else.

It is versatile - one piece of css code powers all instances - just define CSS once and from that point just copy and paste html instance wherever your code requires it.

Every button can have its own label - great for multi-lingual pages (easier that doing pictures for every language - I think) - also allows to propagate instances all over your script easier.

Adjusts its width to label length - also takes fixed width if it is how you want it.
IE7 is an exception to above - it must have width, or will make this button from edge to edge - alternatively to giving it width, you can float button left
- css for IE7:
a. .width:150px; (make note of dot before property, I usually target IE7 by adding such dot - remove dot and property will be read by all browsers)
b. text-align:center; - if you have fixed width, you have to have this to center text/label
c. cursor:pointer; - all IE must have this to show link pointer correctly - good browsers do not need it

You can go step forward with this code and use CSS3 to style it, instead of using images:
a. radius for round corners (limitation: IE will show them square)
b. gradient to make it "button like" (limitation: opera does not support gradients, so remember to set standard background colour for this browser)
c. use :hover pclass to change button states depending on mouse pointer position etc. - you can apply it to text label only, or whole button

CSS code below:

.button_surrogate span { margin:0; display:block; height:25px; text-align:center; cursor:pointer; .width:150px; background:url(left_button_edge.png) left top no-repeat; }

.button_surrogate span span { display:block; padding:0 14px; height:25px; background:url(right_button_edge.png) right top no-repeat; }

.button_surrogate span span span { display:block; overflow:hidden; padding:5px 0 0 0; background:url(button_connector.png) left top repeat-x; }

HTML code below (button instance):

<a href="#">
  <span class="button_surrogate">
     <span><span><span>YOUR_BUTTON_LABEL</span></span></span>
  </span>
</a>