First off, see here from the CRS User's Guide for the definition of ODA.
If you are interested in the aid dependence of recipient countries, visit WDI Online to download "Official development assistance and official aid (current US$)" and "GDP (current US$)" to calculate the amount of foreign aid as the percentage of GDP, as in Erik Werker, Faisal Ahmed, and Charles Cohen (2008) "How is Foreign Aid Spent? Evidence from a Natural Experiment".
If you need more detailed datasets for ODA, visit International Development Statistics Online, by the Development Assistance Committee (DAC) of the OECD.
There are two databases. The DAC database reports the actual disbursement at the aggregate level while the CRS database reports the commitment by sector. See Celasun and Walliser (2008) for the difference between the actual disbursement and the commitment.
Recently the OECD changed the way it distributes these two databases online, and its website is horribly confusing. Basically, they integrate the two databases and the other OECD databases into what they call OECD.STAT. Once you reach the data on the screen, ignore the column on the left. This is for choosing other OECD datasets.
To obtain the total amount of ODA by recipient by year, choose the DAC database on the gateway webpage, and click "DAC 2a". This takes to a data table extracted from OECD.STAT. Click "Recipient" above to narrow the list of recipient countries. Click "Year" above to include data from years before 2000. Then choose "ODA: Total Net" from the drop-down menu for "Aid type". Choose "Constant Prices" for "Amount type". (If you choose aid type and amount type before selecting recipients and years, the aid type and amount type will be reset.) Then click the Excel icon to download the data.
The CRS database reports the amount of ODA committed by sector (education, health, etc.). The list of sectors is available here (an Excel file linked from this page). It may look intimidating to figure out how to download the dataset. Choose the CRS database on the gateway webpage, and find and click "Click here to access the CRS online." This takes to a data table extracted from OECD.STAT. Click "Sector" above to select the sectors of your interest (you can select more than one sector; you'll be able to choose the sector to browse from the drop-down menu for Sector below). Click "Amount" to choose constant values rather than current ones. And click "Year" to select years before 2002 (the earliest is 1974). Click the second icon from the left to make the table rows for recipients rather than for donors. You cannot obtain the total ODA committed from both bilateral and multilateral donors (how inconvenient!). Extract the table for each by choosing "Bilateral" or "Multilateral" from the drop-down menu for "Donor", and then click the Excel icon to download.
See this PDF document for what "channel" refers to.
For how to handle observations with zero or negative values of net ODA, see Levy-Yeyati, Eduardo, Ugo Panizza, and Ernesto Stein, ‘‘The Cyclical Nature of North-South FDI Flows,’’ Journal of International Money and Finance, 26 (2007), 104–130. (Cited in footnote 12 of Hodler and Raskey (2014)).
Used by Easterly, Levine, and Roodman (2004) to check the robustness of the findings by Burnside and Dollar (2000).
Used also by Clemens, Radelet, and Bhavnani (2004) to construct time-series data on "short-impact", "long-impact", and "humanitarian" aid by recipient country and year since 1973.
Used by Stromberg (2007) for which country disbursed disaster relief to which country.
Foreign aid to the health sector
See this post for additional data on foreign aid to the health sector.
Effective Development Assistance
Created by Chang, Fernandez-Arias, and Serven (1998), takes into consideration the difference between grants and consessional loans in the measurement of foreign aid.
Used by Burnside and Dollar (2000).
U.S. economic/military foreign aid from 1946-2001
See Kuziemko and Werker (2006)