In your example, ids is being sent as a part of the body in a POST request but the API expects it to be in the query and the request to be GET.
Try the curl command below:
curl  -u "{username}":"{password}" "https://gateway.watsonplatform.net/concept-insights/api/v2/corpora/public/ibmresearcher/conceptual_search?ids=%5B%22%2Fgraphs%2Fwikipedia%2Fen-20120601%2Fconcepts%2FArtificial_intelligence%22%2C%20%22%2Fgraphs%2Fwikipedia%2Fen-20120601%2Fconcepts%HTML%22%5D&cursor=0&limit=10"
It will do a conceptual search using two concepts Artificial Intelligence and HTML.
Output:
{
  "query_concepts": [{
    "id": "/graphs/wikipedia/en-20120601/concepts/Artificial_intelligence",
    "label": "Artificial intelligence"
  }, {
    "id": "/graphs/wikipedia/en-20120601/concepts/Machine_learning",
    "label": "Machine learning"
  }],
  "results": [{
    "explanation_tags": [{
      "concept": {
        "id": "/graphs/wikipedia/en-20120601/concepts/Machine_Learning",
        "label": "Machine Learning"
      },
      "score": 0.99816346,
      "parts_index": 0,
      "text_index": [
        1024,
        1089
      ]
    }, {
      "concept": {
        "id": "/graphs/wikipedia/en-20120601/concepts/Machine_Intelligence",
        "label": "Machine Intelligence"
      },
      "score": 0.9945005,
      "parts_index": 0,
      "text_index": [
        2097,
        2117
      ]
    }, {
      "concept": {
        "id": "/graphs/wikipedia/en-20120601/concepts/Artificial_intelligences",
        "label": "Artificial intelligences"
      },
      "score": 0.9945005,
      "parts_index": 0,
      "text_index": [
        2557,
        2580
      ]
    }, {
      "concept": {
        "id": "/graphs/wikipedia/en-20120601/concepts/International_Conference_on_Machine_Learning",
        "label": "International Conference on Machine Learning"
      },
      "score": 0.9817866,
      "parts_index": 0,
      "text_index": [
        2658,
        2712
      ]
    }, {
      "concept": {
        "id": "/graphs/wikipedia/en-20120601/concepts/ICML",
        "label": "ICML"
      },
      "score": 0.9817866,
      "parts_index": 0,
      "text_index": [
        2714,
        2718
      ]
    }, {
      "concept": {
        "id": "/graphs/wikipedia/en-20120601/concepts/Feature_selection",
        "label": "Feature selection"
      },
      "score": 0.97459584,
      "parts_index": 1,
      "text_index": [
        1521,
        1538
      ]
    }],
    "id": "/corpora/public/ibmresearcher/documents/il-NOAMS",
    "label": "Slonim, Noam",
    "score": 0.99739265
  }]
}
You need to url encode the json array so for the concepts below:
["/graphs/wikipedia/en-20120601/concepts/Artificial_intelligence", "/graphs/wikipedia/en-20120601/concepts/Machine_learning"]
Using an online URL encoder like http://meyerweb.com/eric/tools/dencoder/
%5B%22%2Fgraphs%2Fwikipedia%2Fen-20120601%2Fconcepts%2FArtificial_intelligence%22%2C%20%22%2Fgraphs%2Fwikipedia%2Fen-20120601%2Fconcepts%HTML%22%5D
Finally use ids=<encoded array> as part of the URL.