A brute force attack against a cipher consists of breaking a cipher by trying all possible keys. Statistically, if the keys were originally chosen randomly, the plaintext will become available after about half of the possible keys are tried. The underlying assumption is, of course, that the cipher is known. Since A. Kerckoffs first published it, a fundamental maxim of cryptography has been that security must reside only in the key. As Claude E. Shannon said a few decades later, 'the enemy knows the system'. In practice, it has been excellent advice.
As of the year 2002, symmetric ciphers with keys 64 bits or fewer are vulnerable to brute force attacks. DES, a well respected symmetric algorithm which uses 56bit keys, was broken by an EFF project in the late '90s. They even wrote a book about their exploit  Cracking DES, O'Reilly and Assoc. The EFF is a nonprofit cyberspace civil rights group; many people feel that wellfunded organisations like the NSA can successfully attack using brute force a symmetric key cipher with a 64bit key. This is surely true, as it has been done publicly. Many observers suggest a minimum key length for symmetric key algorithms of 128 bits, and even then it is important to select a secure algorithm. For instance, many algorithms can be reduced in effective keylength until it is computationally feasable to launch a brute force attack. AES is recommended for use until at least 2010.
The situation with regard to asymmetric algorithms is much more complicated and depends on the individual algorithm. Thus the currently breakable key length for the RSA algorithm is at least 512 bits (has been done publicly), but for most elliptic curve asymmetric algorithms, the largest currently breakable key length is believed to be rather shorter, perhaps as little as 128 bits or so. A message encrypted with a 109 bit key by an elliptic curve encryption algorithm was publicly broken by brute force key search in early 2003. At this writing, 128 bit key lengths seem reasonable for elliptic curve algorithms, and 1024 bits for such other asymmetric key algorithms as RSA (asymmetric key algorithms that rely on complex mathematical problems for their security always will need much larger keyspaces as there are shortcuts to cracking them, as opposed to direct bruteforce).
The term "Brute Force Attacks" is really an umbrella term for all attacks that exhaustively search through all possible (or likely) combinations, or any derivative thereof.
A DictionaryAttack is a common password cracking technique, relying largely on the weak passwords selected by average computer users. For instance, if an attacker had somehow accessed the hashed password files through various malicious database manipulations and educated searching on an online store, he would then write a program to hash one at a time all words in a dictionary (of, for example any or all languages and common derivative passwords), and compare these hashes to the real password hashes he had obtained. If the hashes match, he has obtained a password.
The simple dictionary attack method quickly becomes far too timeconsuming with any large number of password hashes, such as an online database would yeild. Thus, attackers developed the method of precomputation. In this attack, the attacker has already hashed his entire suite of dictionaries, and all he need do is compare the hashes. Additionally, his task is made easier by the fact that many users will select the same passwords. To prevent this attack, a database administrator must attach unique 32bit salts to the users passwords before hashing, thus rendering precompution useless.
