The quickest way for a company to be acquired is the "One Step" method. In this case, the bidder simply calls for a shareholder vote. If the shareholders approve the terms of the offer, the deal can go forward (excepting any legal or other impediments to the deal).
In the "Two-Step" method, which is the case with Dell, the bidder issues a "tender offer" which you mentioned, where the current shareholders can agree to sell their shares to the bidder, usually at a premium. If the bidder secures the acceptance of 90% of the shares, they can immediately go forward with what is called a "short form" merger, and can effect the merger without ever calling for a shareholder meeting or vote. Any stockholders that hold out and do not want to sell are "squeezed out" once the merger has been effected, but retain the right to redeem their outstanding shares at the valuation of the tender offer.
If shareholders controlling say 25% of the shares (not necessarily 25% of the shareholders) were to oppose the tender offer, there would be serveral alternatives. If the bidder did not have at least 51% of the shares secured, they would likely either increase the valuation of the tender offer, or choose to abandon the takeover. If the bidder had 51% or more of the shares secured, but not 90%, they could issue a proxy statement, call for a shareholder meeting and a vote to effect the merger. Or, they could increase the tender offer in order to try to secure 90% of the shares in order to effect the short form merger. If the bidder is able to secure even 51% of the shares, either through the proxy or by way of a controlling interest along with a consortium of other shareholders, they are able to effect the merger and squeeze out the remaining shareholders at the price of the tender offer (majority rules!).
