The chemical composition of horizontal branch (HB) stars might help to clarify the formation history of individual globular clusters (GCs). We studied the Na-O anti-correlation from moderately high resolution spectra for 91 stars on the bimodal HB of NGC 1851; in addition we observed 13 stars on the lower red giant branch (RGB). In our HB sample, 35 stars are on the blue HB (BHB), one is an RR Lyrae, and 55 stars are on the red HB (RHB). The ratio of BHB to RHB stars is close to the total in the cluster (35 and 54%, respectively), while RR Lyrae variables are under-represented, (they are similar to 12% of the NGC 1851 stars). We also derived abundances for He and N in BHB stars. For RHB stars we derived Ba abundances and a few interesting upper limits for N. The RHB stars clearly separate into two groups: the vast majority are O-rich and Na-poor, while about 10-15% are Na-rich and moderately O-poor. Most (but not all) Na-rich RHB stars are also Ba-rich and there is an overall correlation between Na and Ba abundances within the RHB. The group of Ba-rich RHB stars resides on the warmer edge and includes similar to 10% of the RHB stars. We propose that they are the descendant of the stars on the RGB sequence with very red v -y colour. This sequence is known also to consist of Ba and perhaps CNO-rich stars and consistently includes similar to 5-10% of the RGB stars of NGC 1851. However, the upper limit we obtain for N ([ N/ Fe] < 1.55) for one of the Ba-rich stars coupled with the low C-abundances for RGB Ba-rich stars from the literature suggests that the total CNO might not be particularly high ([(C+ N+ O)/ Fe] = 0.15). The other Na-rich RHB stars are also at the warm edge of the RHB and the only RR Lyrae is Na-rich and moderately O-poor. We also find a Na-O anticorrelation among BHB stars, partially overlapping that found among RHB stars, though generally BHB stars are more Na-rich and O-poor. However, there is no clear correlation between temperature and Na and O abundances within the BHB. The average He abundance in BHB stars is Y = 0.29 +/- 0.05, which excludes a large population of extremely He-rich stars from our sample. N abundances are quite uniform at [N/Fe] = 1.16 +/- 0.14 among BHB stars, with a small trend with temperature. This value is consistent with normal CNO abundance and excludes that BHB stars are very CNO-rich: this leaves an age spread of similar to 1.5 Gyr as the only viable explanation for the split of the SGB. To help clarifying the formation history of NGC 1851, we computed synthetic HB's trying to identify which HB stars are the descendant of the bright and faint subgiant branch (b-SGB and f-SGB) stars identified by Milone et al. (2008, ApJ, 673, 241), with respectively 2/3 and 1/3 of the stars of NGC 1851. While most BHB stars likely descend from f-SGB stars and are older, and most RHB stars from b-SGB ones and are younger, the correspondence is probably not one-to-one. In particular, the Ba-rich RHB stars should be less massive than the remaining RHB stars, and the location of their progenitors on the SGB is uncertain. If they descend from f-SGB stars, number counts then require that RR Lyrae variables and possibly some mild BHB stars descend from b-SGB stars; this suggestion is supported by a few circumstantial facts. An investigation of the composition of a large enough sample of SGB stars is required to firmly establish these relations.