How does [0] and [3] wơrk in ASN1?

It sounds like you need an introduction to ASN.1 tagging. There are two angles to approach this from. X.690 defines BER/CER/DER encoding rules. As such, it answers the question of how tags are encoded. X.680 defines ASN.1 itself. As such, it defines the syntax and rules for tagging. Both specifications can be found on the ITU-T website. I'll give you a quick overview.

Tags are used in BER/DER/CER to identify types. They are especially useful for distinguishing the components of a SEQUENCE and the alternatives of a CHOICE.

A tag combines a tag class and a tag number. The tag classes are UNIVERSAL, APPLICATION, PRIVATE, and CONTEXT-SPECIFIC. The UNIVERSAL class is basically used for the built-in types. APPLICATION is typically used for user-defined types. CONTEXT-SPECIFIC is typically used for the components inside constructed types (SEQUENCE, CHOICE, SEQUENCE OF). Syntactically, when tags are specified in an ASN.1 module, they are written inside brackets: [ tag_class tag_number ]; for CONTEXT-SPECIFIC, the tag_class is omitted. Thus, [APPLICATION 10] or [0].

While every ASN.1 type has an associated tag, syntactically, there is also the "TaggedType", which is used by an ASN.1 author to specify the tag to encode a type with. Basically, a TaggedType puts a tag prefix ahead of a type. For example:

MyType ::= SEQUENCE {
  field_with_tagged_type [0] UTF8String
}

The tag in a TaggedType is either explicit or implicit. If explicit, this means that I want the original tag to be explicitly encoded. If implicit, this means I am happy to have only the tag that I specified be encoded. In the explicit case, the BER encoding results in a nested TLV (tag-length-value): the outer tag ([0] in the example above), the length, and then another TLV as the value. In the example, this inner TLV would have a tag of [UNIVERSAL 12] for the UTF8String.

Whether the tag is explicit or implicit depends upon how you write the tag and the tagging environment. For example:

MyType2 ::= SEQUENCE {
  field_with_explicit_tag [0] EXPLICIT UTF8String OPTIONAL,
  field_with_implicit_tag [1] IMPLICIT UTF8String OPTIONAL,
  field_with_tag [2] UTF8String OPTIONAL
}

If you specify neither IMPLICIT nor EXPLICIT, there are some rules that define whether the tag is explicit or implicit (see X.680 31). These rules take into consideration the tagging environment defined for the ASN.1 module. The ASN.1 module may specify the tagging environment as IMPLICIT TAGS, EXPLICIT TAGS, or AUTOMATIC TAGS. Roughly speaking, if you don't specify IMPLICIT or EXPLICIT for a tag, the tag will be explicit if the tagging environment is EXPLICIT and implicit if the tagging environment is IMPLICIT or AUTOMATIC. An automatic tagging environment is basically the same as an IMPLICIT tagging environment, except that unique tags are automatically assigned for members of SEQUENCE and CHOICE types.

Note that in the above example, the three components of MyType2 are all optional. In BER/CER/DER, a decoder will know what component is present based on the encoded tag (which obviously better be unique).

ASN.1 BER and DER use ASN.1 TAGS to unambiguously identify certain components in an encoded stream. There are 4 classes of ASN.1 tags: UNIVERSAL, APPICATION, PRIVATE, and context-specific. The [0] is a context-specific tag since there is no tag class keword in front of it. UNIVERSAL is reserved for built-in types in ASN.1. Most often you see context specific tags to eliminate potential ambiguity in a SEQUENCE which contains OPTIONAL elements. If you know you are receiving two items that are not optional, one after the other, you know which is which even if their tags are the same. However, if the first one is optional, the two must have different tags, or you would not be able to tell which one you had received if only one was present in the encoding.

Most often today, ASN.1 specification use "AUTOMATIC TAGS" so that you don't have to worry about this kind of disambiguation in messages since components of SEQUENCE, SET and CHOICE will automatically get context specific tags starting with [0], [1], [2], etc. for each component.

You can find more information on ASN.1 tags at http://www.oss.com/asn1/resources/books-whitepapers-pubs/asn1-books.html where two free downloadable books are available.

Another excellent resource is http://asn1-playground.oss.com where you can try variations of ASN.1 specifications with different tags in an online compiler and encoder/decoder. There you can see the effects of tag changes on encodings.