RNAi is a tool that employs transfection of siRNAs into a cell. dsRNA is molecule made up of two complementary siRNA or miRNA strnds. dsRNA is acted upon by RISC to generate siRNA or miRNA or miRNA*. all of them are reported to regulate gene expression at post transcriptional level.

RNAi is the name of the process (RNA interference). The RNA interference is a molecular process where an antisenseRNA bind to a complementary target mRNA and drive the mRNA cleavage and degradation. This cause a gene expression silencing at a non-transcriptional but translational level. The action is mediated in the end by a nucleoproteic complex made by a single strand short antisense RNA and the RISC complex (RNA interference silencing complex). So RNAi is a generic name for this kind of non-genomic gene silencing mediated by different RNA species. It's the name of the effect, not the name of the effector.

There are different kind of RNAs that can activate DICER/RISC complexes and trigger gene expression at translational level through RNAi mechanism and this is the reason why you can find long-dsRNA, miRNA, siRNA. All these RNAs finally work as a 19-20 nucleotides single strand antisense RNA attached to the RISC complex for sensing and triggering the right mRNA but they're different at the begin.  

The main difference is between miRNA and siRNA because the siRNA is a double strand short RNA, they're mainly exogenous, for example a viral interference RNA that can aim the host gene regulation, but it can also can be designed experimentally and produced by in vitro transcription for knockdown experiments. Instead the miRNA is a single stranded RNA which fold himself after the synthesis into the nucleus by RNA pol. This stem-loop structure made by the same single strand RNA is cleaved at the level of the loop generating a more or less long double strand RNA derived by an initial single stranded RNA. It is then exported into the cytosol, cleaved and loaded into the same RISC complex. So miRNA is an endogenous alternative gene silencing process, there are a lot of known miRNA expressed in certain situations by cells for a fine gene-regulation.

The long dsRNA is the name of a long (200-500 nt) double strand RNA which is synthetized as a double strand RNA through in vitro transcription and then experimentally delivered into hosts cells for knockdown purposes. Once it is into the cells this long RNA is cleaved by DICER complex exactly like siRNA in many 19-20 nt fragments and these fragments are loaded into the RISC complex for sensing mRNA population searching for the target mRNA.

Not much to add to Paolo's answer.

What else could be said is that dsRNA can be used to define any molecule made up of two complementary strand of RNA.  dsRNA can be formed by two separate complementary strands [like for instance Poly I:C that is known to interact with toll-like receptor 3 (TLR3), which is expressed in the membrane of B-cells, macrophages and dendritic cells. Poly I:C is structurally similar to double-stranded RNA, which is present in some viruses and is a "natural" stimulant of TLR3. Thus, Poly I:C can be considered a synthetic analog of double-stranded RNA and is a common tool for scientific research on the immune system.]  You can also found intra-molecular dsRNA when some long RNA has complementary sequences allowing the long RNA to form secondary structures in the form of a dsRNA.  Not all dsRNA will be a good substrate for DICER complex.

siRNA is usually restricted to SMALL [19 to 25 nt] synthetic fully complementary dsRNA in which one strand is complementary to a specific part of some target mRNA.  And such siRNA will be used to silence expression of the target mRNA after integration into the RISC complex.

Also ALL siRNA is dsRNA when not all dsRNA is siRNA.

Please have a look at attached recent review that may give you more insight. 

I would just like to add a minor correction to Paolo's description, "...siRNA is a double stranded short RNA, it is exogenous...", as siRNA can be endogenous as well. A better differentiation between miRNA & siRNA is, the former is a single continuous chain (resulting in a loop to create double strand) while most other small RNA are single interfering RNA (siRNA).

Neither is necessarily endogenous or exogenous, unfortunately, for ease of differentiation. 

You're right but widely most of them are injected/transfected by us or viral transmitted. Instead, different miRNAs are widely expressed in almost all the cell types in mammals for example. They're clearly the endogenous best example of RNA interference gene silencing. By the way thanks for adding your revision.

Large ds RNA : double stranded RNA formed when complemantary DNA strands are transcribed into RNA ( symmetrical transcription from opposing promoters , these single stranded complementary RNAs base pair to form ds RNA ) , also by base pairing of complementary ssRNAs result from transposons and repetitive genes and viruses . Viral infection of a cell can supply dsRNAs, as many viruses form RNAs of both sense and antisense polarity during replication of their genomes.

siRNAs : exogenous double-stranded RNA ( taken up by cells or it enters through vectors, such as viruses ) of 21-22 nucleotide in length , form through cleavage of long dsRNA by dicer enzyme . siRNAs silence genes by promoting the cleavage of mRNA, with exactly complementary sequences, or recruiting inhibitory proteins, or directing the modification of DNAs with exactly complementary sequences( this occurs in worms and plants but not in animals ) .

miRNA: indigenously produced , short ( 19-25 nt) , single stranded RNAs that are encoded by specific genes in multicellular organisms. They are processed from long single stranded RNA that fold into hairpin structure. They function in repressing mRNA translation ( in case of imperfect complementarity ) or in mRNA degradation ( in case of perfect complementarity ) in plants and animals. In contrast, miRNA can inhibit translation of many different mRNA sequences( perhaps hundreds per miRNA) because its pairing is imperfect .

shRNA : an RNA with tight hairpin turn that can used to silence target gene expression via RNA interference . shRNA is not naturally occurring as miRNA. shRNAs synthesized within the cell by DNA vector-mediated production ( in experiments). They consist of two complementary 19–22 bp RNA sequences linked by a short loop of 4–11 nt similar to the hairpin found in naturally occurring miRNA . Following transcription, the shRNA sequence is exported to the cytosol where it is recognized by Dicer enzyme, which processes the shRNA into the siRNA duplexes. Then this siRNA binds to the target mRNA and is incorporated into the RISC complex for target-specific mRNA degradation .