Hi, all those methods are well described in any books for  molecular biology methods! Just make a little search in the net.

Good luck!

TA-cloning takes advantage of many polymerases' behavior of always leaving an extra 3' A at the end of each strand that it has synthesized (N.B. some proof-reading polymerases remove this overhang). This means that if you have a vector that has been digested in a way such that there are 3' T overhangs at the ends, the DNA fragments will have complementary ends, allowing for bi-directional, efficient ligation of the fragment of interest into the prepared vector.

The Gateway system uses a proprietary set of sequence cassettes that take advantage of the lambda phage recombination system to make many expression vectors from a single donor vector (e.g. you would like to express a gene of interest with many promoters). Each cassette is bounded by attachment ("att") sites, which allow for the integration of sequences flanked by given att sites into others - see documentation for the proper combinations. Once incorporated into an entry vector using the BP clonase mix, the cassette can be moved to the various expression vectors that have different backbone structures, based on your needs. This is accomplished using the LR clonase reaction, and the gene is ready for expression. This is a very effective method for producing many constructs without needing to worry about frame-shifts with each new cloning reaction.

Traditional molecular cloning is mediated by restriction enzymes to cut your vector and fragments so that complementary ends can be ligated together. Much like the TA-cloning, which relies on complementary T and A overhangs, compatible restriction enzymes (for instance, a sequence cut with BamHI will produce an overhang that is compatible with an DpnII-digested sequence - by looking at their cleavage sequences you can see the compatibility of their ends. This allows for directional cloning, which TA-cloning does not provide. For instance, if you find that after TA-cloning, all your transformants are sequenced to show that your insert is oriented 3’ -> 5’ (this can happen due to DNA-repair processes sometimes, that one orientation is more common than another), then directional cloning will ensure that your insert is integrated in the right direction. This is accomplished by linearizing your vector with two different restriction enzymes that allow for different ends that will, in turn, require complementary ends on your insert according to the desired orientation. Blunt-end cloning is another method of cloning, where the ends of both the insert and the vector have been cut with enzymes that leave no overhang, or blunted through other enzymatic means (google “DNA blunting”), though two issues can arise. There is lack of directionality of the insert - only a problem if you worry about your orientation -, and the DNA fragments may recircularize before having the chance to bind to each other, requiring phosphatase treatment before ligation is attempted.

As noted by other answers, looking at the manuals of any kit will give a solid introduction to the theory of each method, and a quick search will give you far more information.

1. TA cloning:

• -A-[PCR fragment]-A-   +  -T-[plasmid backbone]-T-
• (1) With certain DNA polymerases (such as Taq DNA polymerase),   PCR products will have an extra non-template-based 'A' overhang at the both ends
• (2) Plasmid backbone with 'T' overhangs can be purchased from companies ("TA cloning kit")

2. Cloning of PCR products with new restriction ends

• -RS1-[PCR fragment]-RS2-   +  -RS1-[plasmid backbone]-RS2-
• (1) Two primers are designed to carry 2 desirable restriction sites (RS1 and RS2 respectively) and used for PCR amplification, and the resulting PCR product should carry these restriction sites at the ends of the PCR product for later cloning

3. Gateway cloning:

• -L1-[Gene 1]-L2-   x   -R1-[Gene 2]-R2-
• (1) The two DNA fragments (gene 1 and gene 2) swap from 2 plasmids by site-specific recombinations between L1 and R1 (L1 x R1) plus L2 and R2 (L2 x R2) recognition sites, upon the add of a specific site-specific recombinase (see attached figure).

When you amplify your insert with taq polymerase PCR, as a inherent feature of taq, an Adenin extra base leave in the terminals of your insert. By commercially available TA vectors that contain a extra Timine base you can ligate your insert to vector. However, you should remember that the orientation of your insert in this method need furture screening if you are going to clone a protein coding gene.

For more information about Cloning approaches :

1. Article Constructing and transient expression of a gene cassette con...

2. Isolation and bioinformatics study of TbJAMYC transcription factor involved in biosynthesis of taxol from Iranian yew